Math 116 – Matched pairs – Chapter 17 - Show work on another paper – due 11/14/11
The table below shows the weights of seven subjects before and after following a particular diet for two months.
Part 1) Using a 0.01 level of significance, test the claim that the diet is effective in reducing weight.
Part 2) Construct a 98% confidence interval estimate for the mean difference in weights.
BEFORE answering questions, do all of the following:
(1) Can we use t-procedures? Let’s THINK ABOUT assumptions
a) Some important information about HOW the sample was selected is missing. Rephrase the problem to include this important information.
A SIMPLE RANDOM SAMPLE of 7 overweight individuals was selected. Their weights before and after a two-month diet were compared giving the following differences:
List 3 = before – after = 7, 9, 2, -5, 14, -2, 12
b) Since the sample size is small, what are the necessary conditions about the shape of the sample? (check on ROBUSTNESS, chapter 17)
Since the sample is small and there has been no mention about the normality of the population, we have to make sure there are no outliers and no extreme skewness in the sample. In order to do that, read part (c).
c) Checking shape of the “before – after” data
Enter data into L1 and L2 in the calculator, and produce the list L3 = BEFORE – AFTER
Construct a normal probability plot for the BEFORE – AFTER data. Do you get an approximately linear graph?
A normal probability plot for the differences was constructed showing an approximately linear graph. Also, a dot plot and a box plot show no outliers. Since there are no extreme departures from normality we can use the methods learned for normal distributions.
X x x x x x x
__|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|___|
-5 -2 2 7 9 12 14
(2) Use the 1-var-Stats feature to find the relevant statistics of the “before – after” data (which is in L3)
x-bar = d-bar = 5.286, s = 7.158
(3) Now answer question from part (1) - Using a 0.01 level of significance, test the claim that the diet is effective in reducing weight.
· Use the calculator to test the hypothesis. Make sure you
o Write the hypotheses
o Sketch graph, shade and label
o Use a feature of the calculator to obtain the results
o Write the results
o Write conclusion within context.
Ho: mu before = mu after (assume diet makes no difference) mu(d) = 0
Ha: mu before > mu after (diet is effective in reducing weight) mu(d) > 0
Show graph – shade and label
Thinking about the conclusion: p-value = P(d-bar > 5.29) = 0.049 > 0.01 (d-bar is outside of the tail – NEEtoSHa, it’s possible that Ho is correct).
At the 1% significance level, results are not statistically significant. (P = 0.049 > 0.01)
A d-bar of 5.29 or a more extreme one may be observed in 49 out of 1000 samples of the same size selected from the same population of overweight women.
At the 1% significance level, a d-bar of 5.29 or a more extreme one is likely to be observed when there is no difference between the mean weights before and after the diet. At this significance level, the difference observed between d-bar = 5.29 and mu = 0, can be explained by chance variation.
At the 1% significance level there is NOT enough evidence to support the claim that the diet is effective in reducing weight.
(Notice that at the 5% and also at the 10% significance levels the conclusion is different.)
(4) Now answer question from part (2) - Construct a 98% confidence interval estimate for the mean difference in weights.
· Use the calculator to obtain the interval
o Write results
o Explain what the results suggest in reference to the claim that the diet is effective in reducing weight
We are 98% confident that the mean of the differences in weights before and after the diet is between
-3.217 pounds and 13.788 pounds. Since the interval contains zero, and the interval provides plausible values for mu(d), we conclude that it is possible that mu(d) is equal to zero, which implies that the mean weight before the diet may be equal to the mean weight after the diet. With 98% confidence we conclude that the diet is not effective in reducing weight.
(5) Using formulas – Just for fun
· Use formulas to find the test statistic and use the t-table to find the range for the p-value
,
Go with this value to the t-table, under 6 degrees of freedom to read the range for the p-value
0.025 < p-value < 0.05 which means that p > than the significance level alpha of 0.01
· Use formulas to obtain the interval
=
The critical t-value is found from the t-table. Use 6 degrees of freedom and 98% confidence
(6)
· Explain meaning of p-value (read the solution documents from my website for similar interpretations)
P-value = P(d-bar > 5.29) = 0.049 = 49/1000
Refer to part (3)
· Explain the meaning of 98% confidence (read the solution documents from my website for similar interpretations)
o We are 98% confident that the mean of the differences in weights before and after the diet is between -3.217 pounds and 13.788 pounds.
o 98% refers to the success rate of the method used.
o About 98% of the intervals constructed with this method based on samples of the same size, will contain the population mean difference mu that we are estimating.
o If we constructed one-hundred 98% confidence intervals based on one-hundred d-bars from one hundred samples of 7 individuals from the same population, we expect about 98 intervals to contain the population mean mu(d) and about 2 intervals to miss it.
o We are not sure if our interval contains mu(d) but we are 98% confident that it does.
WRONG STATEMENTS:
o 98% of the possible population means mu(d) will be in the interval
o 98% of the possible sample means d-bars will be in the interval
o 98% of the time, our interval (-3.217, 13.788) will contain mu(d)
o There is a 98% probability that our interval (-3.217, 13.788) will contain mu(d)
o There is a 98% probability that d-bar is inside the interval (d-bar IS ALWAYS IN THE CONSTRUCTED INTERVAL, remember that we construct the interval by doing d-bar + or - margin of error)