1. For a random sample of 36 data pairs, the sample mean of the differences was 0.71. The sample standard deviation of the differences was 2. At the 5% level of significance, test the claim that the population mean of the differences is different from 0.

(a) Is it appropriate to use a Student's t distribution for the sample test statistic? Explain.

No, the standard deviation is not smaller than the sample mean.

Yes, the standard deviation is larger than the sample mean.

Yes, the sample size is larger than 30.

No, the sample size is not larger than 30.

What degrees of freedom are used? (b) State the hypotheses.

H₀: _d = 0; H₁: _d > 0

H₀: _d 0; H₁: _d = 0

H₀: _d = 0; H₁: _d < 0

H₀: _d = 0; H₁: _d 0

(c) Compute the t value. (Round your answer to three decimal places.) (d) Estimate the P-value of the sample test statistic.

P-value > 0.500

0.250 < P-value < 0.500

0.100 < P-value < 0.250

0.050 < P-value < 0.100

0.010 < P-value < 0.050

P-value < 0.010

(e) Do we reject or fail to reject the null hypothesis? Explain.

At the = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(f) What do your results tell you?

Fail to reject the null hypothesis, there is sufficient evidence to conclude that the population mean of the differences is not zero.

Reject the null hypothesis, there is insufficient evidence to conclude that the population mean of the differences is not zero.

Fail to reject the null hypothesis, there is insufficient evidence to conclude that the population mean of the differences is not zero.

Reject the null hypothesis, there is sufficient evidence to conclude that the population mean of the differences is not zero.

2.This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise

For a random sample of 36 data pairs, the sample mean of the differences was 1.16. The sample standard deviation of the differences was 3. At the 5% level of significance, test the claim that the population mean of the differences is different from 0.

(a) Is it appropriate to use a Student's t distribution for the sample test statistic? Explain. What degrees of freedom are used? (b) State the hypotheses. (c) Compute the t value. (d) Estimate the P-value of the sample test statistic. (e) Do we reject or fail to reject the null hypothesis? Explain. (f) What do your results tell you?

3.A random sample of 49 measurements from one population had a sample mean of 14, with sample standard deviation 3. An independent random sample of 64 measurements from a second population had a sample mean of 16, with sample standard deviation 4. Test the claim that the population means are different. Use level of significance 0.01.

(a) What distribution does the sample test statistic follow? Explain.

The Student's t. We assume that both population distributions are approximately normal with known standard deviations.

The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.

The standard normal. We assume that both population distributions are approximately normal with known standard deviations.

The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.

(b) State the hypotheses.

H₀: ₁ = ₂; H₁: _{1 2}

H₀: _{1 2}; H₁: ₁ = ₂

H₀: ₁ = ₂; H₁: ₁ > ₂

H₀: ₁ = ₂; H₁: ₁ < ₂

(c) Compute

x₁ x₂.

x₁ x₂ =

Compute the corresponding sample distribution value. (Test the difference _{1 2}. Round your answer to three decimal places.) (d) Estimate the P-value of the sample test statistic.

P-value > 0.500

0.250 < P-value < 0.500

0.100 < P-value < 0.250

0.050 < P-value < 0.100

0.010 < P-value < 0.050

P-value < 0.010

(e) Conclude the test.

At the = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant.

At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

At the = 0.01 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the = 0.01 level, we reject the null hypothesis and conclude the data are not statistically significant.

(f) Interpret the results.

Fail to reject the null hypothesis, there is sufficient evidence that there is a difference between the population means.

Reject the null hypothesis, there is sufficient evidence that there is a difference between the population means.

Reject the null hypothesis, there is insufficient evidence that there is a difference between the population means.

Fail to reject the null hypothesis, there is insufficient evidence that there is a difference between the population means.

(g) Find a 95% confidence interval for

_{1 2}.

(Round your answers to two decimal places.)

lower limit
upper limit

Explain the meaning of the confidence interval in the context of the problem. Because the interval contains only positive numbers, this indicates that at the 95% confidence level,

₁ > ₂.

Because the interval contains both positive and negative numbers, this indicates that at the 95% confidence level,

₁ = ₂.

We can not make any conclusions using this confidence interval.Because the interval contains only negative numbers, this indicates that at the 95% confidence level,

₁ < ₂.

4.

For one binomial experiment,

n₁= 75

binomial trials produced

r₁=45

successes. For a second independent binomial experiment,

n₂= 100

binomial trials produced

r₂=65

successes.At the 5% level of significance, test the claim that the probabilities of success for the two binomial experiments differ.

(a)

Compute the pooled probability of success for the two experiments. (Round your answer to three decimal places.)

(b)

Check Requirements: What distribution does the sample test statistic follow? Explain.

The standard normal. We assume the population distributions are approximately normal.

The standard normal. The number of trials is sufficiently large.

The Student'st. The number of trials is sufficiently large.

The Student'st. We assume the population distributions are approximately normal.

(c)

State the hypotheses.

H₀:p₁=p₂;H₁:p₁<p₂

H₀:p₁=p₂;H₁:p₁>p₂

H₀:p₁<p₂;H₁:p₁=p₂

H₀:p₁=p₂;H₁:p₁p₂

(d)

Computep₁p₂.

p₁p₂=

Compute the corresponding sample distribution value. (Test the differencep₁p₂. Do not use rounded values. Round your final answer to two decimal places.)

(e)

Find theP-value of the sample test statistic. (Round your answer to four decimal places.)

(f)

Conclude the test.

At the= 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.

At the= 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.

At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

At the= 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.

(g)

Interpret the results.

Reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.

Fail to reject the null hypothesis, there is insufficient evidence that the probabilities of success for the two binomial experiments differ.

Reject the null hypothesis, there is insufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.

Fail to reject the null hypothesis, there is sufficient evidence that the proportion of the probabilities of success for the two binomial experiments differ.