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 150 150 200 210 100 110 240 220 200 190 180 150 190 200 360 300 280 260 300 240 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.) Part (a) State the null hypothesis. ○ Ho: Md>0 ○ Ho Hd <0 ○ Ho: Hα = 0 ○ Ho: Ma≥0 Part (b) State the alternative hypothesis. ○ Hai Hd>0 ○ H₂ Hd 20 Part (c) In words, state what your random variable X represents. ○ X represents the difference in the average cholesterol level before and after the diet. OX represents the total difference in cholesterol levels before and after the diet. ○ X represents the average cholesterol level of the 10 individuals. ○ X represents the average difference in the cholesterol level before and after the diet. 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 answer to two decimal places, and if using the t distribution round your answer 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 Ho is false, then there is a chance equal to the p-value that the sample average difference between ending and starting cholesterol levels is less than -9. If Ho is true, then there is a chance equal to the p-value that the sample average difference between ending and starting cholesterol levels is less than -9. If Ho is true, then there is a chance equal to the p-value that the sample average difference between ending and starting cholesterol levels is at most -9. If Ho is false, then there is a chance equal to the p-value that the sample average difference between ending and starting cholesterol levels is at most -9. Part (g) Sketch a picture of this situation. Label and scale the horizontal axis and shade the region(s) corresponding to the p-value. Part (h) О 1/2(p-value) 1/2(p-value) Χα p-value Χα Л. Χα 1/2(p-value) 1/2(p-value) p-value Χα 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.) (ii) Decision: O reject the null hypothesis O do not reject the null hypothesis (iii) Reason for decision: ○ Since p-value a, we reject the null hypothesis. ○ Since p-value > a, we do not reject the null hypothesis. (iv) Conclusion: There is sufficient evidence to show that the cholesterol levels were significantly lowered on the low-fat diet. There is not sufficient evidence to show that the cholesterol levels were significantly lowered on the low-fat diet. Part (i) Explain how you determined which distribution to use. The standard normal distribution will be used because the samples involve the difference in proportions. O The standard normal distribution will be used because the samples are independent and the population standard deviation is known. O The t-distribution will be used because the samples are dependent. ○ The t-distribution will be used because the samples are independent and the population standard deviation is not known.