Question

PLEASE SHOW YOUR WORK CLEARLY.

Obesity in adult males is associated with lower levels of sex hormone. A study investigated a possible link between obesity and plasma testosterone concentrations in adolescent males between the ages of 14 and 20 years. Plasma testosterone level is measured in nanomoles per liter of blood (nmol/l) for a sample of 25 obese adolescent males. The sample mean is 0.26 nmol/l, and the sample standard deviation is 0.11 nmol/l.

a) What distribution does the mean testosterone level in the sample of 25 follow? Specify. This is also known as the sampling distribution of sample means.

           

b) According to the experiment, the single value that can serve as a “best guess” on the mean testosterone level in the population is ________. This is single value is also the point estimate of the population mean.

c) Estimate the population mean testosterone level with an interval (a, b) with a confidence level of 95%, i.e., the 95% confidence interval, which is an interval estimate of the population mean.

Answer 1

Let X be the plasma testosterone concentrations in obese adolescent males between the ages of 14 and 20 year. Let indicate the sample mean for a randomly selected sample of n=25 obese adolescent males.

We know the following from a sample of n=25

The sample mean is nmol/l,

the sample standard deviation is nmol/l.

a) What distribution does the mean testosterone level in the sample of 25 follow? Specify. This is also known as the sampling distribution of sample means.

The sample size is less than 30 and we do not know the population standard deviation. Assuming a normal distribution for the plasma testosterone concentrations in the population, we can say that the sampling distribution of sample mean is t.

ans: the mean testosterone level in the sample of 25 follows a t distribution with degrees of freedom n-1=25-1=24

b) According to the experiment, the single value that can serve as a “best guess” on the mean testosterone level in the population is ________. This is single value is also the point estimate of the population mean.

The sample mean is a good (unbiased) estimate of the population mean. Here, the sample mean is 0.26, which can be used as an estimate of mean testosterone level in the population.

ans: According to the experiment, the single value that can serve as a “best guess” on the mean testosterone level in the population is 0.26 nmol/l

c) Estimate the population mean testosterone level with an interval (a, b) with a confidence level of 95%, i.e., the 95% confidence interval, which is an interval estimate of the population mean.

Since we do not know the population standard deviation, we will use the sample to estimate the same

The estimated population standard deviation is

The standard error of sample mean is

95% confidence interval indicates a significance level

The right tail critical value can be found using

The degrees of freedom for t is n-1=25-1=24

Using the t table, for df=24 and area under the right tail=0.025 (or the combined area = 0.5) we get

We can now get the 95% confidence interval using

ans:  the population mean testosterone level with a confidence level of 95% lies in the interval (0.2146, 0.3054) nmol/l.