# Hypothesis Testing

• The concept of hypothesis testing
• Type 1 and 2 errors
• Blueprint for solving hypothesis test

If you are new to hypothesis testing I'll suggest that you read from start of the page. It covers all the basics and should get you ready for working with test fast.

If you already know about testing you can go to the videos where we demonstrate how tests works.  In these short 5 min tutorials we'll apply hypothesis testing on various questions, a bit like AP exam questions. I'll use use the statlearn calculator that you can download for free here. The point is to give you practical experience with testing and show how the statlearn calculator makes makes your calculations a lot easier.

# Concept ﻿﻿﻿﻿of Hypothesis﻿﻿ ﻿﻿﻿testing

The point of a hypothesis test is to verify a given hypothesis. Specifically, a hypothesis test is seen as a struggle between two conflicting hypotheses, or more precisely, two contradictory hypotheses.

### Definition of a hypothesis

A hypothesis can be seen as an assumption that is not based on facts, but a subjective estimate.

One hypothesis could be, for example, a statement that people are less likely to buy new cars as a result of a financial crisis. It is an assumption that initially seems plausible, but will remain just an assumption until demonstrated by data.

A hypothesis can be seen as an assumption that is not based on facts, but a subjective estimate.

One hypothesis could be, for example, a statement that people are less likely to buy new cars asa result of the ongoing financial crisis. It is an assumption that initially seems plausible, but will remain just an assumption until demonstrated by data.

The starting point for a hypothesis test is defining two conflicting hypotheses: H0 and H1.

The H0 hypothesis represents what is likely to be the current or “true” statement until proven otherwise. The H1 hypothesis is, however, our presumption, drawn up based on a point estimate from a sample. A hypothesis test is intended to determine whether H1 or H0 is true

A hypothesis test can be compared to a trial in which evidence has been collected to suggest that a particular person is guilty (H1). The basis of the trial, however, is that the accused is innocent until proven guilty (H0) The purpose of the trial is, like the purpose of a hypothesis test, to implement a process that results in a verdict: guilty (H1 accepted) or innocent (H0 accepted).

# Defining the null Hypothesis

Suppose you were a sports coach responsible for selecting long jumpers to participate in the Olympics. To participate in the team, a long jumper must have evidence that their average jump is longer than 5 meters.

You are convinced that Max Johnson, who is the team’s best long jumper, can jump more than 5 meters on average. But, since this has not yet been proven, you must assume that the length of his average jump is 5 meters at most. Therefore, you want to perform a hypothesis test to prove that he jumps longer than 5 meters on average and therefore qualifies for the Olympics.

As a coach, you have not recorded the length of each leap Max has taken and you cannot calculate the exact value of his average jump, which corresponds to the population average (µ). Instead, you one day make an attempt to measure the length of 10 of his jumps (n = 10) and calculate the point estimate to be 6.5 meters (Ẍ = 6,5).

This seems promising, but as we said in the section on confidence intervals, the value of a point estimate can vary markedly compared to the population average, especially with small samples.

When we know there may be a relatively large spread in the distribution of point estimates (Ẍ) we cannot exclude the idea that the population average (µ) may be less than 5 meters, despite the fact that the point estimate is 6.5 meters

Due to the variation in the value of the point estimates illustrated, it is possible to obtain point estimates of 6.5 meters (x) despite the average population being only 4 meters (m). On the other hand, it seems intuitive that the larger the distance between Ho and the point estimate, the more it suggests that Ho may be unlikely and the Ho hypothesis is thus incorrect.

From the distribution, we can see that a point estimate of 10 meters is only marginally likely if population average is 5 meters (Ho). In other words, this indicates that, with a point estimate of 10 meters, Max probably has an average jump in excess of 5 meters (H1).

This raises the question: How much bigger than 5 meters should the point estimate be (x) for us to conclude that H1 is true and long jumper Max thereby jumps longer than 5 meters on average? The question brings us to the essence of the hypothesis test: the test level.

The test level is the area of the distribution where Ho is rejected, also known as the “critical area.”

Note: The critical area represents a test level of 5% and has been colored black.

The test level’s size depends on how confident you have to be in order to avoid an erroneous conclusion, or in other words, avoid rejecting H1 when it is true.

Selecting a test level of 5% means there is a 5% chance that the few observations, or point estimates, will end up in the critical area where Ho really is true. To put it another way, there will be a 95% chance that they don’t end up in the critical area if Ho is true. If we have an observation that lies in the critical area, the test concludes, then Ho can be, with a 95% probability, rejected.

Let’s return to the example of Max, the long jumper. We want to investigate whether his average jump is longer than 5 meters. The starting point must necessarily be lower than 5 meters, at least until proven otherwise:

Ho : µ ≤ 5 m and H1 : µ > 5 m

Based on a sample of 10 jumps, his average jump was 6.5 meters (x), with a standard deviation of 2 meters. The question now is whether the 6.5 meters lies within or outside of the critical area.

A calculated hypothesis test result will have a p-value of 2%. The p-value is calculated based on our sample data and can be interpreted as the point estimate of 6.5 meters (x), converted to a scale which is comparable with the test level and indicated as a percentage.

A p-value of 2% means that the point estimate, 6.5 meters, is located in the outer 2% of the distribution, where Ho must be rejected.

Since our test level is 5%, the p-value is within the critical area, which means Ho must be rejected. So, we can, with 95% probability, conclude that the true average jump (m) for Max is longer than 5 meters

# How to calculate Type 1 and 2 errors

Although we have reached a conclusion for this hypothesis test, we can’t be 100% sure that it is right, because there will always be a probability of making an error, which depends on the size of the test level. Selecting a test level of 5% means that there is a 5% chance that we will reject Ho when it is actually true. This is formally known as a type 1 error:

Type 1 error = α = P(reject Ho I Ho true)

Apart from the type 1 error, there is a different and more subtle error which is, not surprisingly, called a type 2 error:

Type 2 error = β = P(accept Ho I Ho false)

As you probably have noticed, type 1 and type 2 errors are based on conditional probabilities, i.e., probabilities that are based on the condition that Ho is either true (type 1 error) or false (type 2 error)

As mentioned, all hypothesis tests are associated with a probability of type 1 and 2 errors. Type 1 errors are governed exclusively by the size of the test level. If the test level is 5%, the probability of a type 1 error corresponds to 5%. So, why not reduce the test level to 0, thus eliminating the likelihood of a type 1 error?

### Example of a type 2 error

The chef at the Hotel D’Angleterre is in a hurry and wants to avoid being disturbed when cooking for the restaurant. Unfortunately, the restaurant has a fire alarm that often rings without cause. To stop the interruptions, you buy a new alarm which, according to the warranty, only has a probability of 0.00001% of ringing when there is no fire.

In this example, we generally assume that there is no fire, until proven otherwise. In this context we can define Ho as “no fire” and H1 as “fire”.

A false alarm corresponds to a type 1 error, since the Ho hypothesis is rejected, despite the fact that it is true. In other words, the erroneous conclusion is that there is a fire when there is not.

If a type 2 error occurs, the situation is obviously worse. Here, we accept that there is no fire, despite the fact that there is one.

In this example, we considered two types of errors, one annoying and the other deadly. It is not always true that type 2 errors are more important than type 1 errors—it depends on the situation. Just remember that minimizing the likelihood of getting one type of error increases the probability of the other type of error occurring.

In contrast to a type 1 error, which is solely determined by the test level, a type 2 error is linked, in theory, to an infinite number of values of H1. Therefore, type 2 errors are presented with a power function showing the probability of committing a type 2 error for varying values of H1.

# How to create a type 2 error function

The owner of a large apple orchard knows, from experience, that his apples weigh an average of 100 grams, with a standard deviation of 5 grams. In a sample of 25 apples, the average weight was 97 grams. In light of this information, consider the following hypotheses:

Ho : µ ≥ 100 grams (the average weight of an apple is more than 100 grams)

H1 : µ ≥ 100 grams (the average weight of an apple is less than 100 grams)

This test requires a calculation of the power function, i.e., the probability of a type 2 error (1 - β). This power function depends on the population parameter being tested and the nature of the hypotheses—see pages 152 and 153 to learn how to calculate the strength of type 2 errors.

In this example, we have the tested population average and H1 < H0. Thus, we come to the following power function:

Here, µ1 can be interpreted for varying values of H1. For simplicity’s sake, we have calculated the power function below using only three possible values of H1 (97, 98, 99). The power function must first be understood as the probability of rejecting H0 for each of the three values of H1.

### Example calculation

Type error function

The power function “for 1-β” speaks for itself—the further the value of H1 (µ1) moves from the value of the hypothesis (µ0), the more likely we are to reject Ho.

From the power function, we can deduce, for example, that if the average weight of the apples for the whole sample is 97 grams, the probability of rejecting Ho will be about 90%.

This example of an application of the power function was based on an average. A similar calculation can be made by testing proportions.

# Summary of Hypothesis﻿﻿ ﻿﻿﻿testing

The purpose of a hypothesis test is to test two opposing hypotheses, Ho and H1. The Ho hypothesis represents our experience thus far. So, we must assume that it is true. The H1 hypothesis is our assumption based on a sample. A hypothesis test is intended to determine whether H1 is true or false.

Ho may be accepted or rejected based on a selected test level. This test level can be interpreted as the critical limit, since the point estimate (in the form of a p-value) must exceed it before we can reject Ho.

There may be two types of errors when a hypothesis test is used. A type 1 error represents the probability of rejecting Ho if it is true. A type 2 error represents the probability of accepting H0 if it is false.

A type 1 error depends solely on the test level’s size. If we use a test level of 5%, the probability of a type 1 error is also 5%.

A type 2 error cannot be tied to a specific value, so this error is instead illustrated with a power function for all possible values of H1.

Vidoes of hypothesis testing with 1 sample

Before watching the videos you can download the blueprints for selecting the right test here

### Testing average, variance known, with statlearn calculator

Hypothesis testing with 2 samples

Hypothesis testing using two samples is based on the same approach as hypothesis testing using a single sample. The only difference is that we compare the two population parameters, such as whether the average return is higher for stock A than B.

To find the correct test for two parameters, use the following table. Please note that, just like with the confidence intervals for the difference between two population parameters, this can lead to different testing options. To test proportions and intensity, there is only one type of test. Therefore, this is not included in the overview.

Videos of hypothesis testing 2 samples

Before watching the videos you can download the blueprints for selecting the right test here

### Testing two proportions

Analysis of variance (ANOVA)

Analysis of variance is a statistical method used to evaluate whether there are differences between the average values across different groups (populations).

An example is a study of how satisfied customers of different banks are. Here, ANOVA would be used to asses whether there is a uniform level of satisfaction, or if there is evidence that customers of some banks are more satisfied than customers of others.

While the above example deals with a quantitative variable (level of satisfaction), ANOVA can also be used to analyze qualitative variables, such as whether used car prices for a particular brand are influenced by that car’s color, upholstery, etc. Thus, ANOVA is considered a flexible method that does not necessarily require quantitative variables, unlike regression analysis.

In the simplest case, where only two populations are compared, ANOVA is like a hypothesis test of the difference between two averages. However, in contrast to performing a hypothesis test of two populations, ANOVA can also be used to test several averages at the same time. The ANOVA effect is assessed based on a single response variable.

For example, a food company may be interested in determining whether there are differences in consumer preferences for five new products that have not yet been launched. In this situation, the company could survey five different test groups (samples). In each survey, respondents _could rate products from 1 to 10 (response variable). Then, the food company could average the results of each of the five samples.

In the above example, ANOVA helps determine whether the variability between the averages of the five samples is high enough (meaning statistically significant) to conclude that the five averages can’t be identical. If the five averages can’t be identical, then we can’t assume all five products are equally popular.

Next, the natural step would be to analyze which products survey respondents preferred. Answering this question is outside of the power of the ANOVA method, and would instead require us to use Tukey’s test. ANOVA just tells us whether the averages we are comparing may be similar.

ANOVA is based on the minimum square method, which is also the basis for regression analysis. To learn more, read the section on the minimum square method, starting on page 184.

Exercises﻿﻿ (﻿﻿﻿with ﻿﻿﻿video ﻿solutions﻿)

The first problem set tests your comprehension of the terms covered in this chapter.

For the first problem set, try to explain the significance of various hypothesis testing terms in your own words. The second problem set requires you to make calculations.

Interpretation exercises:

• What is the purpose of a hypothesis test?
• What is the difference between a hypothesis test and a confidence interval?
• How would you explain the difference between the H0 and H1 hypotheses?
• How would you explain the concept of a test level?
• What effect does the size of the test level have on the conclusion?
• When do you know you are dealing with a test involving two variables?

Calculation exercises:

Problem 1:

Before preparing the marketing strategy for a new type of mortgage loan for Barclays, you need to form an impression of how consumers will perceive the home loan. Therefore, you and the bank have initiated a study in which consumers from different age groups assess whether they think the loan is a good idea. The results are shown below:

Question 1. Determine, using a test level of 1%, whether over half of the customers think that the new mortgage is a good idea.

Question 2. Determine, using a test level of 5%, if there is a difference in the proportion of customers between the ages of 41 and 60 and the customers over 60 who believe that the loan is a good idea. What does this mean for the two segments?

Problem 2:

The rapid sales growth that Green Climate has experienced has brought the company new problems. A growing number of customers are starting to complain about product defects in the company’s ventilation systems. To get an overview of the error rate, a sample of 193 ventilation systems were taken, which included 10 defective units.

Question 1. Calculate the number of defective ventilation units using a 95% confidence level.

Question 2. Determine, using a test level of 5%, whether the defect rate has a cap of 5%.

In order to respond to customers’ complaints, Green Climate has increased the number of quality controls employed in the production process and now guarantees that they will provide customers a better quality product.

One month after Green Climate’s statement regarding their stricter quality controls, a sample of 236 ventilation units found 11 defects.

Question 3. Test whether the increased quality controls have proven successful and resulted in fewer errors, using a test level of 5%.

Problem 3:

It’s almost New Year’s Eve and you have to throw a big party. In a newspaper ad, you spot a bargain from a local fireworks dealer—but because the fireworks are so cheap, you are afraid that there will be a lot of duds in the type of package being sold.

The dealer promises you that fewer than 10% of the fireworks in the package will be duds, and allows you to take one package home and test it yourself. Out of the 35 fireworks you test, 7 are duds.

Is the fireworks dealer ripping you off? Use a test level of 5% when comparing the fireworks you tested with the claims of the fireworks dealer.

Problem 4:

You are developing a new netbook for Acer. To get a sense of what customers will think of the new netbook, you asked product testers what they thought of the prototype version. The test group was made up of 230 people. The results were the as follows:

130 people liked the new netbook a lot.

26 people felt neutral about the new netbook.

74 people disliked the new netbook.

Determine, using a test level of 5%, whether more than half of the product testers liked the new netbook a lot.​

Problem 5:

Recently, the conveyor belt used at the baggage carousel at Copenhagen Airport has been get- ting stuck too often, leading to complaints from disgruntled passengers. To solve the problem, the airport staff has replaced the conveyor belt. Before being replaced, the conveyor belt got stuck 4 times per hour.

After replacing the conveyor belt, the airport staff monitored the baggage carousel over the course of a day. The conveyor belt only got stuck 1.5 times per hour, on average. The Copenhagen Airport staff now assumes that the conveyor belt gets stuck fewer than twice per hour.

Is the assumption of the Copenhagen Airport staff correct? Determine this using a test level of 5%.

Problem 6:

As an employee of MegaFon, you are conducting a poll to dermine whether people of different political affiliations support eliminating early retirement at the same rate. The results are shown below:

Determine, using a test level of 5%, whether red voters approve of the elimination of early retirement. Additionally, comment on the effects of the chosen test level on the final conclusion.

Problem 7:

You work for a major market research agency that is performing market research for the Danish Financial Supervisory Authority. The Danish Financial Supervisory Authority wants to know if there is a link between where urban Danes live and how much confidence they have in the financial sector.

To find out, you survey 150 residents of Aarhus and 200 residents of Copenhagen. Among the residents of Aarhus, 104 say they feel confident about the financial sector. Among the residents of Copenhagen, 132 report feeling confident about the financial sector.

Is confidence in the financial sector greater among the residents of Aarhus than those of Copenhagen? Determine this using a test level of 5%.

Problem 8:

Apple has just developed a new version of the iPad. It is your job to determine the average battery life of this new product. After testing 32 new iPads, you have determined that the average battery life of this device is 7.9 hours, with a standard deviation of 1.8 hours.

Meanwhile, Asus has developed a competing product that uses e-paper. You tested 35 of the new ASUS devices and determined that, on average, this product has an average battery life of 8.1 hours, with a standard deviation of 2.2 hours.

Does the new ASUS e-paper device have a longer battery life than Apple’s latest iPad?

Problem 9:

You’ve set out to examine how much Danish women earn compared to Danish men. The results of the income survey you administered are shown below:

Determine, using a test level of 1%, whether the average income level is higher for men than it is for women.

Problem 10:

McDonald’s is currently expanding throughout China, opening up 2,500 new franchise restaurants. As part of its expansion, McDonald’s wants to assess whether it may be able to benefit from economies of scale by buying potato plantations for their chips, or whether it would be more profitable to import them.

McDonald’s has estimated that each franchise restaurant must buy at least 3 crates of potatoes, weighing 500 kg each, per week for buying a potato plantation to be worthwhile. During a period of 10 weeks, weekly observations were obtained from various franchise restaurants. The average weekly consumption was estimated at 3.9 crates per restaurant.

Question 1. Define the variable and select the right distribution.

Question 2. Does the data suggest that McDonald’s should buy a potato plantation?

Question 3. How significant a change in the data would there need to be for us to come to a different conclusion in Question 2?

In order to not complicate the logistical work involved in transporting potatoes from the plantation to the restaurants, each restaurant must have a weekly consumption of no more than 3 crates, not to be exceeded by more than 15%. Based on the 85 observed restaurants, it was found that there were, at most, 16 that did not use 3 crates a week.

Question 4. Test whether the proportion of stores that did not use 3 crates a week is more than 15%.

Question 5. What difference would it have made if each restaurant had been told that its weekly consumption should be at least 1,242 kg?

Problem 11:

The leading manufacturer of corporate IT systems, SAP, has introduced a new project management system to make it easier to implement its systems and finish projects on time.

Before, 43% of projects were completed within given time frames. However, with the new system, 39 out of 60 projects were completed on time.

Question 1. Define the variable and select the right distribution.

Question 2. Does the new information give us reason to believe that the new project management system works?

Question 3. How would the test level need to be changed to come to the opposite conclusion in the second question?

Question 4. If the true proportion of projects that exceed given time frames is 25%, what is the probability that a maximum of 20 out of 120 projects will not be completed on time?

Problem 12:

To test improvements in audio technology for a new MP3 player, Apple set up two test groups. Group A assessed the sound quality of the company’s current MP3 player and group B rated the sound quality of the new MP3 player. The results can be seen the table below. Marks were given on a 10-point scale, with 10 being the highest rating.

Question 1. Determine, using a test level of 5%, whether the average rating from group B was higher than the average rating from group A.

In group B, there were 28 test subjects who gave the MP3 player a rating of at least 7.

Question 2. Determine, using a test level of 10%, whether more than half of the ratings for the new MP3 player were greater than 7.

Problem 13:

As employee of a research agency, you are currently preparing an analysis of how happy Danes are with their lives. You would like to determine whether age affects life satisfaction. To do so, you have asked adults of all ages to rate how satisfied they generally are with their lives on a scale of 1 to 10, then divided them up into three age groups. The results are shown in the table below.

Determine, using a test level of 5%, whether there is a significant difference in the average level of satisfaction experienced among Danes from different age groups.

# Hypothesis Testing Exercises

Hypothesis testing exercises (page and problem numbers):

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Page 10 /1.1

Page 10 /1.3

Page 10 /2.2 (raw data)

Page 10 /2.3 (raw data)

Page 10 /2.4 (summary data combined with raw data)

Page 10 /4.2 (focus on the level of sensitivity)

Page 14 /4.3

Page 27 /2.3 (focus on the level of sensitivity)

# Solutions for Hypothesis Testing Questions

The following shows solutions for the exercises involving hypothesis testing. For the sake of simplicity, the solutions have been reduced down to just the results. For an exam, it is important that you show your work, including the details provided in the section titled “General method for solving hypothesis testing problems.”

Interpretation exercises:

• You can test whether a given claim, in the form of an H1 hypothesis, is true or false.
• A test leads to the conclusion that a population parameter is smaller/larger/different from a certain value presented by an H0 and H1 hypothesis. A confidence interval does not lead to the same kind of conclusion, but instead shows an interval containing a population para- meter within a given probability (usually 95%).
• The H0 hypothesis represents the conditions that we must, for now, assume are true. H0 is, in other words, what we must assume is correct until proven otherwise. The H1 hypothesis challenges the H0 hypothesis with the opposite claim. The H1 hypothesis is made on the basis of a point estimate indicating that the H0 hypothesis is wrong. To find out which of the two opposing hypotheses reflects the truth, a hypothesis test is used.
• The test level is the test tolerance value. The test level represents the probability of com- mitting a type 1 error, which means rejecting a true H0 hypothesis. The lower the test level, the lower the probability of committing a type 1 error. On the other hand, lowering the test level increases the probability of a type 2 error (a type 2 error is accepting an H0 hypothesis when it is false). The test level should not be lowered without weighing the possible effects of an increased likelihood of a type 2 error.
• The larger the test level, the greater the probability of committing a type 1 error, and thus, rejecting an H0 hypothesis that is true.
• When point estimates come from two different samples.

Calculation exercises:

Problem 1.

Question 1.

X: Number of customers who consider the new mortgage a good idea.

X ~ b(p = 0,5835, n = 533) when the variable is discrete and there is independence between the events.

Conclusion.

Since p-value (0,000058) , the test level (0.01), H0 is rejected, meaning that more than half of customers believe that the new mortgage is a good idea.

Question 2.

A: 41 to 60-year-olds who believe that the new mortgage is a good idea.

B: 60+ year-olds who believe that the new mortgage is a good idea.

A ~ b(p = 0,608, n = 181) and B ~ b(p = 0,662, n = 157) when the variable is discrete and there is independence between the events.

Conclusion.

Since the p-value (0.149076) . the test level (0.05), H0 is accepted. Thus, we cannot assume that there is a difference in the proportion of those who consider the mortgage a good idea in these two different age groups.

Problem 2.

Question 1.

X: The number of defective ventilation units.

X ~ b(p, n = 193) when the variable is discrete and there is independence between the events.

95% confidence interval for the proportion

Lower limit: 0,02054277

Upper limit: 0,08308418

Question 2.

X: The number of defective ventilation units.

X ~ b(n = 193, p = 0,0518) when the variable is discrete and there is independence between the events.

Conclusion.

Since the p-value (0.454) > the test level (0.01), H0 is accepted, meaning that the proportion of defective ventilation units can, with a probability of 99%, not be considered more than 5%.

Question 3.

A: The number of defective ventilation units before improvement initiatives.

B: The number of defective ventilation units after improvement initiatives.

A and B ~ b(n, p) when the variable is discrete and there is independence between the events.

Conclusion.

Since the p-value (0.4018) > the test level (0.05), H0 is accepted. Thus, we cannot say that there is a difference in the proportion of defective ventilation units after the implementation of quality improvement initiatives.

Problem 3.

X: Number of duds

X ~ b(p, n = 35 when the variable is discrete and there is independence between the events

Since the p-value (0,0243033) < test level (0.05), we must reject H0.

The fireworks dealer is ripping you off.

Problem 4.

X: Product testers who like the new netbook a lot.

X ~ b(p = 0,5652, n = 230) when the variable is discrete and there is independence between the events.

Since the p-value (0,0239564) < test level (0.05), we must reject H0.

We must therefore assume that more than half of the consumers like the new netbook a lot.

Problem 5.

X: Number of times conveyor belt gets stuck per hour.

X ~ Ps(λ = 1,5)

Since the p-value (0,0416323) < test level (0.05), we must reject H0.

So, the assumption that the Copenhagen Airport has made about the newly-repaired conveyor belt is correct. However, the conclusion is level sensitive, since the p-value is close to the test level.

Problem 6.

X: The number of red voters who support the elimination of early retirement.

X ~ b(p = 0,5692, n = 130) when the variable is discrete and there is independence between the events

Since the p-value (0,0572019) > the test level (0.05), we must accept H0.

We can’t deny that up to half of those who vote on red block, are against eliminating early retirement. However, the conclusion is level sensitive, which means that we would not have to change the test level much to come to the opposite conclusion.

Problem 7.

A: People from Aarhus who have confidence in the financial sector.

B: People from Copenhagen who have confidence in the financial sector.

Approximate z-test of the difference between two proportions.

Since the p-value (0,255104) > the test level (0.05), we must accept H0.

Confidence in the financial sector is not greater among the people of Aarhus than among the people of Copenhagen.

Problem 8.

A: Battery life for new iPad (measured in hours).

B: Battery life for new ASUS e-paper device (measured in hours).

Z-test of the difference between the two averages

Since the p-value (0,3414006) > the test level (0,05) we must accept H0.

Therefore, it cannot be said that the new ASUS e-paper device has a longer battery life than Apple’s new iPad.

Problem 9.

A: Income for men.

B: Income for women.​

Z-test of the difference between the two averages

Since the p-value (0,0009209) < the test level (0.01), we must reject H0.

So, we can conclude that the average income for men is greater than the average income for women.

Problem 10.

Question 1.

X: Number of crates consumed per week per restaurant.

X ~ Ps(λ = 3,9)

Question 2.

(n = 10)

Conclusion: Since the p-value (0.05017) > the test level (0.05), we cannot reject H0.

Question 3.

The p-value would only have to be modified by a few thousandths (from 0.05017 to 0.04999) to lead to the opposite conclusion. Since the p-value is so close to the test level, the conclusion is very level sensitive, meaning that the conclusion reached by using a test level of 5% could actually go either way. Since the H0 hypothesis is not really significantly rejected using a test level of 5%, we could reasonably argue that McDonald’s should still invest in the potato plantation.

Question 4.

Conclusion:

Since the p-value (0.162) > the test level (0.01), we should accept the H0 hypothesis.

Question 5.

In this case, the variable would be continuous. Thus instead of using proportion we should have tested with an average.

Problem 11.

Question 1.

X: Number of projects that exceed their time frames.

X ~ b(p, n = 60) when the variable is discrete and there is independence between events, meaning the various projects. In this case, there is a constant probability that a project will exceed the time frame.​

Question 2.

Conclusion:

Since the p-value (0.105343) > the test level (0.05), H0 is accepted. This means that we can refute the claim that the project management system works.

Question 3.

From 0,05 to 0,11 (i.e., 5% to 11%).

Question 4.

X ~ b(p = 0,25, n = 120)

P(x ≤ 30) = 0,548853​

Problem 12.

Question 1.

F-test of 2 standard deviations (homogeneity testing):

Conclusion:

Since the p-value (0.066) > the test level (0.05), H0 is accepted. This represents the assumed homogeneity of the variance, which means that the variance for the two populations can be assumed to be identical.

Pooled t-test of the difference between 2 means (variance).

Conclusion:

Since the p-value (0.0003984294) < the test level (0.05), H0 is rejected. This means that the average rating can, with a probability of 95%, be considered greater for group B than for group A.

Question 2.

Conclusion:

Since the p-value (0.124) > the test level (0.1), H0 is accepted. This means that a maximum of half of the ratings for the new MP3 player will be greater than 7.

Problem 13.

Data

Assumptions:

Sample has been selected randomly.

Obs. are from normally distributed populations.

Populations have equal variances.

Conclusion:

Since the p-value (0,0451219) < the test level (0.05), we must reject H0.

Conclusion:

There are differences in the average level of satisfaction across age groups when measured at a test level of 5%.

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