Your uncle believes that eating pine needles lowers cholesterol. The table shows the cholesterol levels of four randomly selected individuals before and after trying out a pine needle-based diet. Is there enough evidence to support your uncle’s claim at ??=0.10 ? ?You may assume that cholesterol levels are normally distributed.
a. Identify and calculate bar and sd
c. Use your calculator to identify the ??-value
b. Identify the null and alternative hypotheses.
d. Decide whether to reject or fail to reject 0
e. Interpret your decision in context.
The Correct Answer and Explanation is :
To analyze whether eating pine needles lowers cholesterol levels, we need to perform a hypothesis test. I’ll guide you through each step:
a. Identify and calculate ( \bar{d} ) and ( s_d )
Given that you’ve mentioned having data for individuals before and after the diet, ( \bar{d} ) (the mean difference) and ( s_d ) (the standard deviation of the differences) need to be calculated. Let’s say the data looks something like this:
| Person | Before (mg/dL) | After (mg/dL) | Difference ( d ) |
|---|---|---|---|
| 1 | 210 | 200 | -10 |
| 2 | 220 | 210 | -10 |
| 3 | 240 | 230 | -10 |
| 4 | 230 | 220 | -10 |
First, calculate the mean difference (( \bar{d} )):
[ \bar{d} = \frac{\sum d_i}{n} = \frac{-10 – 10 – 10 – 10}{4} = -10 ]
Then, calculate the standard deviation (( s_d )):
[ s_d = \sqrt{\frac{\sum (d_i – \bar{d})^2}{n-1}} ]
[ s_d = \sqrt{\frac{0 + 0 + 0 + 0}{4 – 1}} = 0 ]
b. Identify the null and alternative hypotheses.
- Null Hypothesis ( H_0 ): Pine needles do not lower cholesterol levels, or the mean difference in cholesterol levels is zero (( \mu_d = 0 )).
- Alternative Hypothesis ( H_a ): Pine needles lower cholesterol levels, or the mean difference in cholesterol levels is less than zero (( \mu_d < 0 )).
c. Use your calculator to identify the ( p )-value
Since the sample size is small (n=4), and assuming normal distribution, you would typically use a t-test to determine the ( p )-value. However, as ( s_d ) is zero (indicating no variability among differences), standard statistical tests like the t-test might not be applicable here because the test statistic involves division by the standard deviation. This could mean all measured differences are exactly the same, indicating a limitation in the variability or a possible error in data collection or recording.
d. Decide whether to reject or fail to reject ( H_0 )
Since ( s_d = 0 ), it poses a challenge to proceed with a standard t-test. We may need to reconsider the test or the approach. However, with the mean difference consistently showing a decrease and if accepting ( s_d ) is practically zero:
- If proceeding formally, you would not typically reject ( H_0 ) because you cannot compute a valid t-statistic.
- Practically, you might argue that the consistent decrease in each individual supports the claim, but statistically, the analysis isn’t robust without variability.
e. Interpret your decision in context.
Given the limitations in variability and potential issues with using ( s_d ) to compute a t-statistic, we cannot statistically reject ( H_0 ) based on the standard methods. Practically, the data suggests a consistent decrease in cholesterol after consuming pine needles, which might support your uncle’s claim. However, scientifically and statistically, it’s recommended to have more varied data or to use different methods to confirm this effect more robustly. The conclusion here hinges critically on the appropriateness of the data and methods for the hypothesis in question.