Question

During fiscal year 2004, the mean monthly benefit per person participating in the federal food-stamp program in Nebraska was $79.52. Due to the poor economy, more people are using this support program to purchase eligible food. A random sample of 15 people using food stamps was obtained, the sample mean was $80.97 and the sample standard deviation was $4.50. Assume the distribution of monthly benefit per person is normal and use α=0.05. Is there any evidence the true mean monthly benefit per person has increased? If it is not possible to answer this question, explain. (a) Formulate the null and alternative hypotheses. (b) Compute the appropriate test statistic. (c) Compute the P-value. (d) State your conclusion (reject or fail to reject the null hypothesis). (e) Interpret your conclusion in context. (f) Find the probability of committing the Type I error. (g) If the true mean monthly benefit per person is $82, find the power of the test and the probability of committing the Type II error.

Answer 1

Sample size = n = 15

Sample mean = = 80.97

Standard deviation = s = 4.50

a) Claim: The true mean monthly benefit per person has increased.

The null and alternative hypothesis is

Level of significance = 0.05

b) Here population standard deviation is unknown so we have to use t-test statistic.
Test statistic is

c) Degrees of freedom = n - 1 = 15 - 1 = 14

P-value = P(T > 1.25) = 0.1163

P-value > 0.05 we fail to reject null hypothesis.

d) Conclusion: The true mean monthly benefit per person has NOT increased.