Question 1

In 2010,the proportion of employees at a large company who commuted to work by car was 8%.A current survey investigates whether the proportion of employees who commute by car to work is higher than it was in 2010.What are the null and alternative hypotheses?

Accepted Answer

The best choice for the null and alternative hypotheses would be:
H0: p = 0.08 (proportion of employees who commuted to work by car in 2010)
HA: p > 0.08 (proportion of employees who commute to work by car now is higher than it was in 2010)
This is because we are testing if the proportion of employees who commute by car now is higher than it was in 2010, which implies that the null hypothesis should be the proportion in 2010 and the alternative hypothesis should be the proportion now being greater. Thus, option (C) is the correct choice.

Question 2

Suzie has installed a new spam blocker program on her email.She used to receive an average of 20 spam emails a day.Is the new program working?

Accepted Answer

The null hypothesis ($\mathrm { H } _ { 0 }$) represents the situation before the change (receiving an average of 20 spam emails a day), and the alternative hypothesis ($\mathrm { H } _ { \mathrm { A } }$) represents the expected outcome if the spam blocker is working (receiving less than 20 spam emails a day).

Question 3

In April of 2011,the unemployment rate in Canada was reported to be 7.6%.Suppose we claim that the current unemployment rate differs from the April 2011 unemployment rate.If we wish to support our claim,what are the null and alternative hypotheses?

Accepted Answer

The null hypothesis (H0) represents the statement that there is no change from the historical or known value, which in this case is the unemployment rate being 7.6% (or 0.076 as a proportion). The alternative hypothesis (HA) represents the claim we wish to support, which is that the current unemployment rate is different from 7.6%. Therefore, HA: p ≠ 0.076 correctly states that the current rate is not equal to 7.6%, making A the correct choice.

Question 4

In April of 2011,the unemployment rate in Canada was reported to be 7.6%.Statistics Canada claims that the current unemployment rate is lower than that of April 2011.To test this claim,a hypothesis test is conducted and yields a P-value turns out to be 0.04.Provide an appropriate conclusion.

Accepted Answer

A P-value of 0.04 indicates that there is strong evidence against the null hypothesis (that the current unemployment rate is the same as that of April 2011) and in favor of the alternative hypothesis (that the current unemployment rate is lower than that of April 2011). Therefore, we can conclude that there is strong evidence that Statistics Canada is correct and the unemployment rate has decreased. Choice E correctly states this interpretation of the P-value.

Question 5

The management of a water park has concerns about the chlorine level of 0.4% mg/mL increasing to an unsafe level.The management tests the hypothesis that the park's chlorine proportions have remained the same,and find a P-value of 0.005.Provide an appropriate conclusion

Accepted Answer

A P-value of 0.005 (or 0.5%) indicates that there is a very low probability of observing the sample results if the null hypothesis (no change in chlorine proportions) were true. This suggests strong evidence against the null hypothesis, implying that the chlorine proportion has likely increased.

Question 6

According to the 2004 Canadian Community Health Survey,23.1% of Canadians aged 18 or older were considered obese; that is,they had a body mass index (BMI)of 30 or higher.Suppose you believe that the percentage of obese Canadians has increased since 2004 and wish to test your belief.Based on a 2014 survey,you conduct a hypothesis test and come up with a P-value of 0.028.What is reasonable to conclude?

Accepted Answer

A P-value of 0.028 means that if the percentage of obese Canadians has not changed since 2004, there is only a 2.8% chance of seeing the increase in the percentage of obese Canadians that we observed or higher from natural sampling variation. This suggests strong evidence that the percentage of obese Canadians has increased since 2004. Therefore, choice D is the best answer.

Question 7

You want to see if the number of minutes cell phone users use each month has changed from its mean of 120 minutes 2 years ago.You take a random sample of 100 cell phone users and find an average of 135 minutes used.

Accepted Answer

The answer of You want to see if the number...

Question 8

After a political ad campaign for a city's mayor,pollsters check the disapproval ratings.They test the hypothesis that the ads produced no change against the alternative that the ratings are now below 19%,and find a P-value of 0.17.Provide an appropriate conclusion.

Accepted Answer

The answer of After a political ad campaign for a...

Question 9

A weight loss program provided a loss for 60% of its participants.The program leader decides to test a new weight loss strategy to see if it's better and gets a P-value of 0.21.Is it reasonable to conclude that the new strategy and the old one are equally effective?

Accepted Answer

The answer of A weight loss program provided a loss...

Question 10

Are Canadians getting fatter? Researchers interested in this question take a random sample of 500 people and record an average weight of 86 kg.Ten years ago,the average weight was 83 kg.

Accepted Answer

We can use a one-tailed hypothesis test with the null hypothesis being that the average weight is still 83 kg (μ=83) and the alternative hypothesis being that the average weight has increased (μ>83). 
Since the sample average weight is 86 kg, which is higher than the null hypothesis value, we can reject the null hypothesis and conclude that the average weight has increased in the past ten years. Out of the given answer choices, only B matches this conclusion.

Question 11

A university wants to increase its retention rate of 4% for graduating students from the previous year.After implementing several new programs during the last two years,the university reevaluates its retention rate and comes up with a P-value of 0.075.What is reasonable to conclude about the new programs?

Accepted Answer

A P-value of 0.075 means that there is a 7.5% chance of seeing the increase in retention rate (or higher) from natural sampling variation, assuming that the new programs have no effect on retention. Therefore, we have suggestive but inconclusive evidence that the new programs are more effective. It is not correct to say that there is a chance of the new programs having an effect, as the P-value does not provide information about causation. Similarly, we cannot conclude that the new programs are more effective, as the evidence is not strong enough. Lastly, it is not correct to say that there is a chance of the new programs having no effect, as the P-value only refers to natural sampling variation and does not provide information about the programs themselves.

Question 12

A city claims that 75% of all low income housing is 1500 sq.ft.The tenants believe the proportion of housing this size is smaller than the claim,and hire an independent firm to test an appropriate hypothesis.What are the null and alternative hypotheses?

Accepted Answer

The null hypothesis should state that the proportion of low income housing that is 1500 sq.ft is equal to 0.75, as that is the claim made by the city. The alternative hypothesis should state that the proportion is less than 0.75, as the tenants believe it is smaller than the claim. Therefore, the correct choice is C.

Question 13

A survey investigation whether the proportion of employees who commute by car to work is higher than it was five years ago finds a P-value of 0.009.Is it reasonable to conclude that more employees are commuting by car?

Accepted Answer

The answer of A survey investigation whether the proportion of...

Question 14

Health Canada reports that,in 2007,85.7% of Canadians were satisfied with the health care services they received.Suppose you are now concerned that a smaller percentage of Canadians are satisfied with the health care services they received.If you want to justify your concerns,what are the null and alternative hypotheses?

Accepted Answer

The null hypothesis should state that there is no significant difference between the current satisfaction rate and the previous satisfaction rate. Therefore, the null hypothesis should be H0: p = 0.857. The alternative hypothesis should state that there is a significant difference between the current and previous satisfaction rates. As we are concerned that the satisfaction rate may have decreased, the alternative hypothesis should be HA: p < 0.857. Therefore, the best choice is C.

Question 15

The seller of a loaded die claims that it will favor the outcome 6.We don't believe that claim,and roll the die 350 times to test an appropriate hypothesis.Our P-value turns out to be 0.01.Provide an appropriate conclusion.

Accepted Answer

The answer of The seller of a loaded die claims...

Question 16

A new manager,hired at a large warehouse,was told to reduce the 26% employee sick leave.The manager introduced a new incentive program for employees with perfect attendance.The manager decides to test the new program to see if it's better and receives a P-value of 0.06.What is reasonable to conclude about the new strategy?

Accepted Answer

A P-value of 0.06 suggests that if there were no effect of the program, there is a 6% chance of observing the data or something more extreme due to random variation. This provides moderate evidence against the null hypothesis, but not strong enough to conclusively say the program is effective.

Question 17

Has the introduction of more music download services changed the number of university students downloading music online without paying for it? Two years ago,approximately 50,000 college students admitted to downloading music online without paying for it.

Accepted Answer

The null hypothesis ($\mathrm { H } _ { 0 }$) typically represents no change or the status quo, so it should state that the mean ($\mu$) number of students downloading music without paying is equal to 50,000, as this was the number two years ago. The alternative hypothesis ($\mathrm { H } _ { \mathrm { A } }$) should reflect any change from this status quo, hence it should state that the mean is not equal to 50,000 ($\mu ≠ 50,000$), indicating a change in behavior due to the introduction of more music download services.

Question 18

A city claims that 75% of all low income housing is 1500 sq.ft.The tenants believe the proportion of housing this size is smaller than the claim,and hire an independent firm to test an appropriate hypothesis.The P-value turns out to be 0.025.Provide an appropriate conclusion.

Accepted Answer

The P-value of 0.025 indicates that, assuming the city's claim is true, there is only a 2.5% chance of observing a sample proportion of 1500 sq.ft. housing as low or lower than what was observed. This provides strong evidence against the city's claim, supporting the tenants' belief that the proportion is less than 75%.

Question 19

You are considering moving to Richmond Hill and are concerned about the average one-way commute time to downtown Toronto.Does the average one-way commute time exceed 25 minutes? You take a random sample of 100 Richmond Hill residents and find an average commute time of 29 minutes with a standard deviation of 7 minutes.

Accepted Answer

The null hypothesis (H0) is that the average one-way commute time to downtown Toronto is equal to or less than 25 minutes. The alternative hypothesis (Ha) is that the average one-way commute time to downtown Toronto is greater than 25 minutes.
Since the sample size (100) is large and the population standard deviation is unknown, we can use a one-sample t-test. We can calculate the t-statistic as:
t = (x̄ - μ) / (s/√n) = (29 - 25) / (7/√100) = 4
Using a significance level of 0.05 and degrees of freedom (df) = n - 1 = 99, the critical t-value for a one-tailed test is 1.660. Since the calculated t-value (4) is greater than the critical t-value (1.660), we reject the null hypothesis and conclude that the average one-way commute time from Richmond Hill to downtown Toronto exceeds 25 minutes. This is represented by the choice B, which states that the average commute time is greater than 25 minutes.

Question 20

Is the mean weight of female college students equal to 59 kg? To test this,you take a random sample of 80 students,finding a mean of 61 kg with a standard deviation of 8.4 kg.

Accepted Answer

The P-value (0.0332) is less than the significance level (0.05), so we reject the null hypothesis.

Quiz 17: Testing Hypotheses About Proportions