ECON 321: CH.9

To test μ for an x distribution that is mound-shaped using sample size n ≥ 30, how do you decide whether to use the normal or Student's t distribution?

If σ is known, use the standard normal distribution. If σ is unknown, use the Student's t distribution with n - 1 degrees of freedom.

If we fail to reject (i.e., "accept") the null hypothesis, does this mean that we have proved it to be true beyond all doubt? Explain your answer.

No, it suggests that the evidence is not sufficient to merit rejecting the null hypothesis.

If we reject the null hypothesis, does this mean that we have proved it to be false beyond all doubt? Explain your answer.

No, the test was conducted with a risk of a type I error.

For the same sample data and null hypothesis, how does the P-value for a two-tailed test of μ compare to that for a one-tailed test?

The P-value for a two-tailed test is twice the P-value for a one-tailed test.

In a statistical test, we have a choice of a left-tailed test, a right-tailed test, or a two-tailed test. Is it the null hypothesis or the alternate hypothesis that determines which type of test is used? Explain your answer.

The alternative hypothesis because it specifies the region of interest for the parameter in question.

To use the normal distribution to test a proportion p, the conditions np > 5 and nq > 5 must be satisfied. Does the value of p come from H0, or is it estimated by using p̂ from the sample?

The value of p comes from H0.

In general, if sample data are such that the null hypothesis is rejected at the α = 1% level of significance based on a two-tailed test, is H0 also rejected at the α = 1% level of significance for a corresponding one-tailed test? Explain your answer.

Yes. If the two-tailed P-value is smaller than α, the one-tailed area is also smaller than α.

A random sample of 51 adult coyotes in a region of northern Minnesota showed the average age to be x = 2.01 years, with sample standard deviation s = 0.76 years. However, it is thought that the overall population mean age of coyotes is μ = 1.75. Do the sample data indicate that coyotes in this region of northern Minnesota tend to live longer than the average of 1.75 years? Use α = 0.01. (a) What is the level of significance? (b)State the null and alternate hypotheses. (c)What sampling distribution will you use? Explain the rationale for your choice of sampling distribution. (d)What is the value of the sample test statistic? (Round your answer to three decimal places.) (e)Estimate the P-value. (f)Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis? Are the data statistically significant at level α? (g)Interpret your conclusion in the context of the application.

a)0.01 b)H0: μ = 1.75 yr; H1: μ > 1.75 yr c)The Student's t, since the sample size is large and σ is unknown. d)2.443 e)P-value < 0.010 f)At the α = 0.01 level, we reject the null hypothesis and conclude the data are statistically significant. g)There is sufficient evidence at the 0.01 level to conclude that coyotes in the specified region tend to live longer than 1.75 years.

(a) What is a null hypothesis H0? (b) What is an alternate hypothesis H1? (c) What is a type I error? (d)What is a type II error? (e) What is the level of significance of a test? (f)What is the probability of a type II error?

a)A working hypothesis making a claim about the population parameter in question. b)Any hypothesis that differs from the original claim being made. c)Type I error is rejecting the null hypothesis when it is true. d)Type II error is failing to reject the null hypothesis when it is false. e)The probability of a type I error. f)β

A random sample of 25 values is drawn from a mound-shaped and symmetric distribution. The sample mean is 15 and the sample standard deviation is 2. Use a level of significance of 0.05 to conduct a two-tailed test of the claim that the population mean is 14.5. (a) Is it appropriate to use a Student's t distribution? Explain. (b)How many degrees of freedom do we use? 24 (c)What are the hypotheses? (d)Compute the t value of the sample test statistic. (Round your answer to three decimal places.) (e)Estimate the P-value for the test. (f)Do we reject or fail to reject H0? (g)Interpret the results.

a)Yes, because the x distribution is mound-shaped and symmetric and σ is unknown b)24 c)H0: μ = 14.5; H1: μ ≠ 14.5 d)t=1.250 e)0.100 < P-value < 0.250 f)At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. g)There is insufficient evidence at the 0.05 level to reject the null hypothesis.

A random sample of 50 binomial trials resulted in 20 successes. Test the claim that the population proportion of successes does not equal 0.50. Use a level of significance of 0.05. (a) Can a normal distribution be used for the p̂ distribution? Explain. (b) State the hypotheses. (c) Compute p̂. (d)Compute the corresponding standardized sample test statistic. (Round your answer to two decimal places.) (e)Find the P-value of the test statistic. (Round your answer to four decimal places.) (f)Do you reject or fail to reject H0?Explain. (g)What do the results tell you?

a)Yes, np and nq are both greater than 5. b)H0: p = 0.5; H1: p ≠ 0.5 c)0.4 d)-1.41 e)0.1586 f)At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant. g)The sample p̂ value based on 50 trials is not sufficiently different from 0.50 to justify rejecting H0 for α = 0.05.

The body weight of a healthy 3-month-old colt should be about μ = 64 kg. (a) If you want to set up a statistical test to challenge the claim that μ = 64 kg, what would you use for the null hypothesis H0? (b) In Nevada, there are many herds of wild horses. Suppose you want to test the claim that the average weight of a wild Nevada colt (3 months old) is less than 64 kg. What would you use for the alternate hypothesis H1? (c) Suppose you want to test the claim that the average weight of such a wild colt is greater than 64 kg. What would you use for the alternate hypothesis? (d) Suppose you want to test the claim that the average weight of such a wild colt is different from 64 kg. What would you use for the alternate hypothesis? (e) For each of the tests in parts (b), (c), and (d), respectively, would the area corresponding to the P-value be on the left, on the right, or on both sides of the mean?

a)μ = 64 kg b)μ < 64 kg c)μ > 64 kg d)μ ≠ 64 kg e)left; right; both

When using the Student's t distribution to test μ, what value do you use for the degrees of freedom?

n - 1

Consider a binomial experiment with n trials and r successes. To construct a test for a proportion p, what value do we use for the z value of the sample test statistic?

r/n