**STAT 250 Homework 4**

Question 1 Background :

Wild Safari

A “wild safari” park allows visitors to drive their cars through over 400 acres to see more than 1000 animals roaming freely. The park claims that the probability of some car damage by an animal during a safari drive-thru is 0.65. Suppose 20 cars are selected at random and 18 cars have some damage. Does it appear that the population proportion of cars with some damage exceeds the park’s claim?

Question 1 Subquestions

1.a

2 point(s)

The hypotheses to be tested are H_{0}: *p* = 0.65 versus H_{a}: *p* > 0.65. Clearly define the parameter *p *in the context of the problem.

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1.b

1 point(s)

Report the observed value of the test statistic for testing the hypotheses in part (a).

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1.c

1 point(s)

If the null hypothesis is true, what is the distribution of the test statistic (that is, which distibution should be used to find the *p*-value)?

· N(0,1) distribution

· N(0.65, 0.11) distribution

· Bin(n=20, p=0.65) distribution

· Bin(n-20, p=0.90) distribution

1.d

2 point(s)

Compute the *p*-value for this test.

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1.e

1 point(s)

Our decision at the 5% level of significance is to reject the null hypothesis. Based on our decision, the appropriate conclusion in context is:

(**C hallenge**

*:*before looking at the four options, try writing out the conclusion in context yourself ~ then compare it to the four options ~ good practice.)

· There is not sufficient evidence to conclude that the sample proportion of cars with some damage exceeds the park’s claim.

· There is not sufficient evidence to conclude that the population proportion of cars with some damage exceeds the park’s claim.

· There is sufficient evidence to conclude that the sample proportion of cars with some damage exceeds the park’s claim.

· There is sufficient evidence to conclude that the population proportion of cars with some damage exceeds the park’s claim.

1.f

1 point(s)

If the study will be repeated in the future with the same sample size selected from the sample populaton, but now a 10% significance level *α* will be used, then the probability of correctly rejecting the null hypothesis would

· increase

· decrease

· stay the same

Question 2 Background :

Bolt Machine A or B?

**Machine A or Machine B?** At the JB (Just Bolts) factory there are two machines operating next to each other. Machine A makes *15 mm* bolts and Machine B makes *14 mm* bolts. Each machine produces bolts whose lengths vary normally around their target mean length with a standard deviation of 0.5 mm.

When a bolt is found on the floor in between these two machines, the operator uses the following decision rule to determine which bin to throw that bolt into: **Reject H**_{0}** (and conclude the bolt was made by Machine B) if the length of the bolt is 14.5 mm or shorter. **

**Hint: **For parts (a), (c), (e), think of what is stated to be the "*truth*" (which machine actually made the bolt), and that will tell you which distribution to work with. And you will want to use the Empirical Rule when possible to find some of the probabilities.

Question 2 Subquestions

2.a

0.5 point(s)

For this situation, what is the probability the operator will **reject H**** _{0}** when

**H**

**is true? That is, what is the probability the operator will throw a bolt actually made by Machine A into the bin for Machine B?**

_{0}· 0.025

· 0.05

· 0.16

· 0.32

· 0.68

· 0.84

· 0.95

· 0.975

2.b

0.5 point(s)

Which of the following correspond to the probability computed in part (a)? Select all that apply.

· alpha

· the significance level

· the probability of a type 1 error

· the probability of a type 2 error

· beta

· power

· p-value

2.c

0.5 point(s)

For this situation, what is the probability the operator will **reject H**** _{0}** when

**H**

**is true? That is, what is the probability the operator will throw a bolt actually made by Machine B into the bin for Machine B?**

_{a}· 0.025

· 0.05

· 0.16

· 0.32

· 0.68

· 0.84

· 0.95

· 0.975

2.d

0.5 point(s)

Which one of the following corresponds to the probability computed in part (c)? Select one.

· alpha

· the significance level

· the probability of a type 1 error

· the probability of a type 2 error

· beta

· power

· p-value

2.e

0.5 point(s)

Suppose the operator has just found a bolt on the floor and the observed length is 14 mm. What is the probability of seeing a bolt as short as 14 mm or even shorter if it was really made by Machine A?

· 0.025

· 0.05

· 0.16

· 0.32

· 0.68

· 0.84

· 0.95

· 0.975

2.f

0.5 point(s)

Which one of the following corresponds to the probability computed in part (e)? Select one.

· alpha

· the significance level

· the probability of a type 1 error

· the probability of a type 2 error

· beta

· power

· p-value

1 point(s)

Question 3 :

Error

The significance level was set at 5% and the decision was to reject H_{0}, what type of mistake could have been made?

Question 3 Multiple Choice Options

- Type 1 error

- Type 2 error

Go to question:

1

2

3

4

5

6

7

8

Question 4 Background :

An Unusual P-Value

Suppose a researcher collected data to assess if a “majority” of all residents of a particular city were in favor of a newly proposed city ordinance. Her hypotheses are H_{0}: *p *= 0.5 versus H_{a}: *p* > 0.5. A large enough random sample of residents was obtained and the resulting sample proportion was 0.41.

Question 4 Subquestions

4.a

1 point(s)

**Think about the test statistic value: **Consider the formula for the z test statistic. Although its exact value cannot be computed without knowing the actual sample size *n, *it can be determined that the resulting test statistic value will be:

· positive

· negative

4.b

1 point(s)

**Think about the p-value: **Consider the sign of the resulting test statistic value and the hypotheses that are to be tested and 'picture' the

*p*-value. (It is recommended that you actually sketch the picture of the

*p*-value -- draw the model for your z statistic assuming the null hypothesis is true, and then shade the area that would be the

*p*-value.) What can you say about that

*p*-value?

· It is equal to 0.05.

· It is less than 0.50.

· It is equal to 0.50.

· It is larger than 0.50.

Question 5 Background :

Comparing Rates for Favoring Bond Proposal

Suppose a bond proposal for school construction will be submitted to voters at the next municipal election. A major portion of the money will be used to build schools in a rapidly “developing” section of the city whereas the remaining money will be used to update current school buildings. To assess the viability of the proposal, a random sample of 50 residents who live in the developing section of the city was selected and an independent random sample of 100 residents who do not live in the developing section of the city was selected. All 150 residents were asked how they plan to vote for the proposal. The results are summarized below.

Let *p*_{1} represent the population proportion of all residents of the developing section that are in favor of the proposal, and *p*_{2} represent the population proportion of all residents from outside the developing section that are in favor of the proposal.

Question 5 Subquestions

5.a

1 point(s)

Estimate the difference in the two population proportions *p*_{1} – *p*_{2}.

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5.b

2 point(s)

Provide the 99% confidence interval estimate for the difference in the population rate of favoring the proposal for all residents of the developing section as compared to all residents from outside the developing section.

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5.c

0.5 point(s)

A 90% confidence interval based on the same data would result in an interval that is ___________ the 99% confidence interval.

· wider than

· narrower than

· the same width as

5.d

1 point(s)

The computation of the 99% confidence interval, which involved using a z* multiplier, requires a check that the sample sizes are large enough. Which of the following is the correct check at that condition?

· The sample sizes of 50 and 100 are both at least 10.

· The sample sizes of 50 and 100 are both at least 30.

· The values of 50, 100, 38, and 65 are all at least 10.

· The values of 38, 12, 65, and 35 are all at least 10.

Question 6 Background :

Job Offer and Time with Family

“Have you ever __refused__ a job, promotion, or transfer because it would lead to spending less time with your family?” This question was asked in a survey of two independent random samples representing two populations:

- Population 1: all working adults living in the East with a family of at least two children
- Population 2: all working adults living in the Midwest with a family of at least two children

The researcher would like to assess if the proportion of all such working adults living in the East who have refused a job for this reason would exceed the same proportion for all such working adults living in the Midwest. The significance level is set to 5%. The results of the study are summarized below.

Question 6 Subquestions

6.a

1 point(s)

The researcher sets up the null hypothesis as H_{0}: *p*_{1} = *p*_{2}. Select the appropriate alternative hypothesis.

· Ha: p1 not equal to p2

· Ha: p1 > p2

· Ha: p1 < p2

6.b

0.5 point(s)

It is important to understand what the various parameters in the hypotheses represent. Consider the following *incorrect *definition for the parameter *p*_{1}.

**“ p**

_{1}

*is the sample proportion of working adults from the East who have refused a job for this reason.***”**Provide the two words to complete this statement: This definition would be correct if you replace the word _____________ with the word _______________.

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6.c

1 point(s)

The test statistic is computed assuming the null hypothesis is true, that is, *p*_{1} = *p*_{2} = *p*. Which of the following is the best estimate of that common proportion* p*?

· 0.1085

· 0.23605

· 0.2376

· 0.4721

6.d

2 point(s)

It seems that the sample sizes of 93 and 88 should be sufficiently large enough to use a normal approximation to compute the *p*-value. Provide verification that the sample sizes are large enough.

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6.e

3 point(s)

The z test statistic value is 1.71. Find the corresponding *p*-value. Include a complete sketch to show the *p*-value. (You may make your own complete sketch by hand and upload image as a .jpg or use either the prob() script or even better pval() script in R to create the picture for you.) Then below the (inserted image) sketch provide the actual value by writing "*p*-value = _________."

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6.f

1 point(s)

At a 5% significance level, the decision is to reject the null hypothesis. Which of the following word(s) should be used to complete the appropriate one sentence conclusion in the context of the problem?

For the populations represented by our samples, there ___________ sufficient evidence to support that the population proportion of all working adults living in the East (with at least two children) who have refused a job for this reason exceeds that for all working adults living in the Midwest (with at least two children).

· IS

· IS NOT

1 point(s)

Question 7 :

What does it really mean? Significance level

At a 5% significance level, researchers plan to conduct a test of the difference of two proportions to compare the rate of alcohol use among college students for two universities. When the study is conducted next Fall, rates of use will be estimated based on large independent random samples of college students from the two universities. Which of the following statements is the best interpretation of the proposed significance level of 5% in this context?

Question 7 Multiple Choice Options

- There is a 5% chance that the rate of alcohol use is different for the two universities.

- There is a 5% chance that the rate of alcohol use is the same for the two universities.
- If the rate of alcohol use is actually the same for the two universities, there is a 5% chance that the researchers will mistakenly conclude the rate of alcohol use is different for the two universities.

- If the rate of alcohol use is not the same for the two universities, there is a 5% chance that the researchers will mistakenly conclude the rate of alcohol use is the same for the two universities.

P-values

Determine whether each of the following statements is **True or False**.

Question 8 Subquestions

8.a

1 point(s)

The *p*-value is the probability the null hypothesis is true.

· True

· False

8.b

1 point(s)

The larger the *p*-value, the more evidence there is to stay with the null hypothesis.

· True

· False

**Category:**Business, General Business

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