Question

Ten individuals went on a low-fat diet for 12 weeks to lower their cholesterol. The data are recorded in the table below. Do you think that their cholesterol levels were significantly lowered? Conduct a hypothesis test at the 5% level.

Starting cholesterol level	Ending cholesterol level
140	140
220	250
110	130
240	220
200	190
180	150
190	200
360	300
280	300
260	240

NOTE: If you are using a Student's t-distribution for the problem, including for paired data, you may assume that the underlying population is normally distributed. (In general, you must first prove that assumption, though.)

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

b.) What is the test statistic? (If using the z distribution round your answer to two decimal places, and if using the t distribution round your answer to three decimal places.)

t = ______

c.) What is the p-value? (Round your answer to four decimal places.)

d.) 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 =  

Answer 1

Since we are given the data for ten individuals at two different time period , we have to perform paired t test

a) t distribution with df = 10 - 1 = 9

So t₉  distribution to use for the test.

Claim : cholesterol levels were significantly lowered

H₀ : µ_d = 0 vs H_a : µ_d > 0  

First we need to find the differences d = Starting cholesterol level - Ending cholesterol level

Then we need to find the mean difference and standard deviation of differences S_d

We can use excel to find d , and S_d

Therefore   = 6 and S_d = 27.5681

To find t statistic and P- value we can use TI-84 calculator .

Press STAT key ---> Scroll to TESTS ---> select T Test and hit enter.

plug the given values accordingly

Scroll to calculate and hit enter.

b) t = 0.688

c) P- value = 0.2543

d) As P- value is greater than α = 0.05

We do not reject the null hypothesis.

Conclusion : There is no significance evidence that the cholesterol levels significantly lowered.

(i) α = 0.05