Question

An article in the San Jose Mercury News stated that students in the California state university system take 4.5 years, on average, to finish their undergraduate degrees. Suppose you believe that the mean time is longer. You conduct a survey of 46 students and obtain a sample mean of 5.1 with a sample standard deviation of 1.2. Do the data support your claim at the 1% level? Note: If you are using a Student's t-distribution for the problem, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

Part (a) State the null hypothesis. H0: μ ≥ 4.5 H0: μ ≠ 4.5 H0: μ < 4.5 H0: μ ≤ 4.5

Part (b) State the alternative hypothesis. Ha: μ ≠ 4.5 Ha: μ = 4.5 Ha: μ > 4.5 Ha: μ ≥ 4.5

Part (c) In words, state what your random variable X represents. X represents the average number of students who receive an undergraduate degree in California. X represents the length of time it takes a student to finish his or her undergraduate degree. X represents the average length of time it takes students to finish their undergraduate degrees. X represents the average number of students in the California university system.

Part (d) State the distribution to use for the test. (Enter your answer in the form z or tdf where df is the degrees of freedom.)

Part (e) What is the test statistic? (If using the z distribution round your answers to two decimal places, and if using the t distribution round your answers to three decimal places.) ---Select--- = Part (f) What is the p-value? (Round your answer to four decimal places.) Explain what the p-value means for this problem. If H0 is true, then there is a chance equal to the p-value the average time needed to complete an undergraduate degree is not 5.1 years or more. If H0 is true, then there is a chance equal to the p-value that the average time needed to complete an undergraduate degree is 5.1 years or more. If H0 is false, then there is a chance equal to the p-value the average time needed to complete an undergraduate degree is not 5.1 years or more. If H0 is false, then there is a chance equal to the p-value that the average time needed to complete an undergraduate degree is 5.1 years or more.

Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot

Part (h) Indicate the correct decision ("reject" or "do not reject" the null hypothesis), the reason for it, and write an appropriate conclusion.

(i) Alpha (Enter an exact number as an integer, fraction, or decimal.) α =

(ii) Decision: reject the null hypothesis do not reject the null hypothesis

(iii) Reason for decision: Since α > p-value, we reject the null hypothesis. Since α < p-value, we do not reject the null hypothesis. Since α < p-value, we reject the null hypothesis. Since α > p-value, we do not reject the null hypothesis.

(iv) Conclusion: There is sufficient evidence to conclude that the average time it takes to finish the undergraduate degrees is longer than 4.5 years. There is not sufficient evidence to conclude that the average time it takes to finish the undergraduate degrees is longer than 4.5 years.

Part (i) Construct a 95% confidence interval for the true mean. Sketch the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval. (Round your lower and upper bounds to two decimal places.) WebAssign Plot

Answer 1