Question 1

Which information will be comprehensive when summarizing the results of a hypothesis test? ​

Accepted Answer

The summary should include the hypotheses being tested, the test procedure used, the test statistic calculated, and the conclusion of the test in the context of the problem. The sample size alone does not provide enough information, and the boxplot and normal probability plot (while potentially useful) are not necessary for a comprehensive summary. Significance level and test power are important aspects of the test, but they are not always calculated or reported and may not be necessary for a basic summary.

Question 2

Do the students in the campus spend less on food than an average American resident does? To answer this question, expenses of 25 students were calculated. ​  ​ Food expenses of an average American person are $220. Interpret the output of the Shiny app "Randomization test for a population mean" using a significance level of 0.05. ​  ​

Accepted Answer

The p-value is 0.024, which is smaller than the significance level of 0.05. This means that we reject the null hypothesis and conclude that there is convincing evidence that students' expenses are less than the national average.

Question 3

City officials will conduct a study of the proportion of vehicle owners among the neighborhood residents. The study, which was held 5 years ago, provided the result of 50%. Suppose that a random sample of size n = 20 with the obtained proportion 0.45 is used to decide if the proportion has increased for the whole neighborhood. Calculate the P-value using the appropriate test to analyze these data. ​

Accepted Answer

With a small sample size of 20, a large-sample z test is not appropriate. Therefore, options A and C are eliminated. Since the null hypothesis is that the proportion has not increased and the alternative hypothesis is that the proportion has increased, a one-tailed test can be used. Both a randomization test for one proportion or an exact binomial test for one proportion can be used in this case, making option D the correct choice. The P-value is between 0.06 and 0.08, based on the calculated test statistic and the appropriate distribution (either the binomial or sampling distribution).

Question 4

The local airport authority wants to investigate whether the mean of handled passengers per month has increased from 64,500 to plan maintenance works. Determine the valid null and alternative hypotheses.

Accepted Answer

The null hypothesis should state that there is no difference or no change in the mean number of handled passengers per month, which is µ = 64,500. The alternative hypothesis should state that there is an increase in the mean number of handled passengers per month, which is represented by µ > 64,500. Therefore, option B is the valid null and alternative hypotheses. Option A, C, D, and E are invalid because they either have incorrect symbols or do not represent the correct hypotheses.

Question 5

A cooler bag producing company claims that their coolers keep products cold for 12 hours. A test of hypotheses H₀: µ = 12 versus Ha: µ < 12 is based on a sample of size n = 33. Suppose that σ is known to be 0.6 and a significance level of 0.05 will be used. Determine the power of the test and interpret the result when µ = 11.75.

Accepted Answer

The power of the test can be calculated using the formula: Power = P(Z < (Zα-μ/σ/√n) - Z(μ/σ/√n-Δ)) where Zα-μ/σ/√n is the critical value for the given significance level (0.05) and Z(μ/σ/√n-Δ) is the standard normal value for the alternative hypothesis mean (11.75) and the null hypothesis mean (12) when σ is known to be 0.6. Substituting the given values, we get: Power = P(Z < -1.64 - (-0.42)) = P(Z < -1.22) = 0.77 Therefore, the power of the test is 0.77, which means that 77% of the samples will result in a conclusion to reject H₀: µ = 12 when the true µ is 11.75.

Question 6

One of the early decisions that must be made when performing a hypothesis test is the choice of significance level. Briefly describe the considerations that go into making this decision.

Accepted Answer

The answer of One of the early decisions that must...

Question 7

Researchers study how the global warming affects the sea level. They wonder whether the mean of yearly differences of the sea level has increased from 4.4 mm using the level of significance α = 0.05. What does the level of significance mean in the context of the study?

Accepted Answer

The level of significance, α = 0.05, represents the probability of rejecting the null hypothesis when it is actually true. In this context, it means there's a 5% chance of concluding that the mean yearly difference in sea level has increased from 4.4 mm when it has not changed.

Question 8

A hypothesis test for a population mean uses t statistic, when the population standard deviation σ is known.

Accepted Answer

A t statistic is used when the population standard deviation σ is unknown. When σ is known, a z statistic is typically used for hypothesis testing.

Question 9

Select the correct statement about the test power.

Accepted Answer

The statement "When H₀ is false, power = 1 - β." is the correct statement about test power. H₀ refers to the null hypothesis being true (HS), the alternative hypothesis being greater than the null (U1), the level of significance being 0.10 (B10), the standard deviation being 1 (S1), and the sample size being 1 (U1). When the null hypothesis is false, the power of the test is the probability of rejecting the null hypothesis when it is false, which is equal to 1 minus the probability of a Type II error (β). Therefore, the correct statement is C) When H₀ is false, power = 1 - β.

Question 10

A journal article summarizes findings of a study on the effect of color on perception of temperature. The article states that experiments involving 32 participants showed an insignificant difference in the effect of orange and violet on subjective feeling of cold. On average, it was 0.3° warmer in the orange room than in the violet one, when participants started to complain about being cold. The sample standard deviation of the differences was s = 0.7. Does this provide convincing evidence that there is no significant difference in the effect of orange and violet on subjective feeling of cold in general? Test the relevant hypotheses using a α = 0.05. ​

Accepted Answer

The answer of A journal article summarizes findings of a...

Question 11

The average wingspan of a crowned eagle is supposed to be 40 in. A random sample of 31 crowned eagles raised in captivity gave the following results:  = 39.5, s = 1.8. Do the data indicate that the wingspan of captive eagles is less than that of those that live in the wild? Use the significance level α = 0.05.

Accepted Answer

The answer of The average wingspan of a crowned eagle...

Question 12

According to a survey conducted 5 years ago, the proportion of left-handed students at a university was 0.12. Has the proportion increased over the years? A random sample of 50 students was chosen, 9 of whom were left-handed. Use the output from the Shiny app "Randomization Test for One Proportion" to complete an appropriate hypothesis test. ​  ​ Interpret the obtained results, using a significance level of 0.1. ​

Accepted Answer

Based on the output from the Shiny app, the p-value for the test is 0.175. This means that, with a significance level of 0.1, there is no convincing evidence to reject the null hypothesis that the proportion of left-handed students has remained the same. Therefore, the best choice is C, "There is no convincing evidence that the proportion of left-handed students has increased."

Question 13

When performing tests of hypotheses, there are assumptions that must be met in order for the test to be appropriate. For the test of a hypothesis about a population proportion, describe how you would check the assumptions.

Accepted Answer

The answer of When performing tests of hypotheses, there are...

Question 14

Determine the P-value in a hypothesis test for a population mean with H₀: µ = hypothesized value and Ha: µ > hypothesized value.

Accepted Answer

When testing a hypothesis for a population mean with a one-tailed alternative hypothesis (in this case, µ > hypothesized value), the p-value is the area to the right of the calculated t statistic under the t distribution with df = n - 1. Therefore, choice C is the correct answer.

Question 15

The university conducts a study designed to find out the proportion of students who have conversational fluency in two or more foreign languages. The previous study provided the result of 7%. The test is carried out using data from a random sample of n = 72. In this sample, the number of students who speak two or more foreign languages is 10. Complete an appropriate hypothesis test with α = 0.05 and make a conclusion using the output from the Shiny app "Exact Binomial Test for One Proportion". ​  ​

Accepted Answer

The answer of The university conducts a study designed to...

Question 16

A company provides portable walkie-talkies to construction crews. Their batteries last, on average, 55 hours of continuous use. The purchasing manager receives a brochure advertising a new brand of batteries with a lower price, but suspects that the lifetime of the batteries may be shorter than the brand currently in use. To test this, the new brand is installed in 8 randomly selected radios. Here are the results for the lifetime of the batteries (in hours): 45 52 56 55 51 57 48 52 Is there sufficient evidence to conclude that the purchasing manager is correct in his conjecture that the new brand has a shorter average lifetime?

Accepted Answer

The answer of A company provides portable walkie-talkies to construction...

Question 17

The average sleeping time required for teenagers who experience major physical exertion is about 8 hours. Do the members of the school sport teams sleep enough? Sleeping time for a random sample of 16 sport team members was obtained. ​  ​ Which test is appropriate for this sample? ​

Accepted Answer

The answer of The average sleeping time required for teenagers...

Quiz 10: Hypothesis Testing Using a Single Sample

Which information will be comprehensive when summarizing the results of a hypothesis test?

The local airport authority wants to investigate whether the mean of handled passengers per month has increased from 64,500 to plan maintenance works. Determine the valid null and alternative hypotheses.

One of the early decisions that must be made when performing a hypothesis test is the choice of significance level. Briefly describe the considerations that go into making this decision.

Researchers study how the global warming affects the sea level. They wonder whether the mean of yearly differences of the sea level has increased from 4.4 mm using the level of significance α = 0.05. What does the level of significance mean in the context of the study?

A hypothesis test for a population mean uses t statistic, when the population standard deviation σ is known.

Select the correct statement about the test power.

When performing tests of hypotheses, there are assumptions that must be met in order for the test to be appropriate. For the test of a hypothesis about a population proportion, describe how you would check the assumptions.

Determine the P-value in a hypothesis test for a population mean with H₀: µ = hypothesized value and Ha: µ > hypothesized value.

The average sleeping time required for teenagers who experience major physical exertion is about 8 hours. Do the members of the school sport teams sleep enough? Sleeping time for a random sample of 16 sport team members was obtained. Which test is appropriate for this sample?

Quiz 10: Hypothesis Testing Using a Single Sample

Which information will be comprehensive when summarizing the results of a hypothesis test? ​

The local airport authority wants to investigate whether the mean of handled passengers per month has increased from 64,500 to plan maintenance works. Determine the valid null and alternative hypotheses.

One of the early decisions that must be made when performing a hypothesis test is the choice of significance level. Briefly describe the considerations that go into making this decision.

Researchers study how the global warming affects the sea level. They wonder whether the mean of yearly differences of the sea level has increased from 4.4 mm using the level of significance α = 0.05. What does the level of significance mean in the context of the study?

A hypothesis test for a population mean uses t statistic, when the population standard deviation σ is known.

Select the correct statement about the test power.

When performing tests of hypotheses, there are assumptions that must be met in order for the test to be appropriate. For the test of a hypothesis about a population proportion, describe how you would check the assumptions.

Determine the P-value in a hypothesis test for a population mean with H0: µ = hypothesized value and Ha: µ > hypothesized value.

The average sleeping time required for teenagers who experience major physical exertion is about 8 hours. Do the members of the school sport teams sleep enough? Sleeping time for a random sample of 16 sport team members was obtained. ​ ​ Which test is appropriate for this sample? ​

Which information will be comprehensive when summarizing the results of a hypothesis test?

Determine the P-value in a hypothesis test for a population mean with H₀: µ = hypothesized value and Ha: µ > hypothesized value.

The average sleeping time required for teenagers who experience major physical exertion is about 8 hours. Do the members of the school sport teams sleep enough? Sleeping time for a random sample of 16 sport team members was obtained. Which test is appropriate for this sample?