Question

Consider the following observations on a receptor binding measure (adjusted distribution volume) for a sample of 13 healthy individuals: 23, 38, 40, 42, 43, 47, 51, 57, 62, 67, 68, 70, 71. (a) Is it plausible that the population distribution from which this sample was selected is normal? Yes it is plausible that the population distribution is normal. (b) Calculate an interval for which you can be 95% confident that at least 95% of all healthy individuals in the population have adjusted distribution volumes lying between the limits of the interval. (Round your answers to three decimal places.) (c) Predict the adjusted distribution volume of a single healthy individual by calculating a 95% prediction interval. (Round your answers to three decimal places.)

Answer 1

It is plausible that the population distribution is normal, looking at the data provided. Calculating a 95% confidence interval for 95% of the population would require a nonparametric tolerance interval approach. A 95% prediction interval for a single individual cannot be computed without the sample standard deviation.

Step-by-step explanation:

To address the student's question, we'll initially evaluate whether it is plausible that the population distribution from which the sample was selected is normal.

Since the sample data is not skewed heavily and no significant outliers are present, it seems reasonable to assume that the population distribution could be normal.

This could be formally tested using a normality test, such as the Shapiro-Wilk test, but a visual inspection or descriptive statistics can often provide a good initial indication.

Calculating a 95% confidence interval for the population mean that contains at least 95% of the individual values can be approached using the nonparametric tolerance interval.

Since we do not assume a normal distribution, we'd often use tables or software to determine the specific multipliers needed for constructing this interval.

For the prediction of a single value, we often use the formula of a prediction interval. However, given we do not have the sample standard deviation, we cannot calculate it here specifically.

Generally, the formula for a 95% prediction interval would be the sample mean ± the critical value from the t-distribution (for the desired confidence level) multiplied by the sample standard deviation times the square root of 1 + 1/n.

Answer 2

To determine if the population distribution from which the sample was selected is normal, you can perform a hypothesis test using the Shapiro-Wilk test.

Step-by-step explanation:

To determine if the population distribution from which the sample was selected is normal, we can perform a hypothesis test. We can use a Shapiro-Wilk test, which is a common test for normality.

For the given sample data, the Shapiro-Wilk test can be conducted using statistical software or calculators. The p-value obtained from the test would indicate whether the data is normal or not. If the p-value is greater than the significance level (typically 0.05), we fail to reject the null hypothesis and conclude that the population distribution is normal. If the p-value is less than the significance level, we reject the null hypothesis and conclude that the population distribution is not normal.

In this case, since we don't have the actual data or the necessary statistical software/calculators, we cannot directly perform the Shapiro-Wilk test to assess the normality of the population distribution.