Question

A researcher is interested in finding a 95% confidence interval for the mean number minutes students are concentrating on their professor during a one hour statistics lecture. the study included 105 students who averaged 36.2 minutes concentrating on their professor during the hour lecture. the standard deviation was 13.1 minutes. round answers to 3 decimal places where possible.

a. to compute the confidence interval use a t correct distribution.
b. with 95% confidence the population mean minutes of concentration is between and minutes.
c. if many groups of 105 randomly selected members are studied, then a different confidence interval would be produced from each group.
d. about percent of these confidence intervals will contain the true population mean minutes of concentration and about percent will not contain the true population mean minutes of concentration.

Answer 1

A 95% confidence interval for the average concentration time can be calculated using the sample mean, standard deviation, and a t-distribution for the given sample size of 105 students. This interval will represent the range where the true population mean is likely to lie, with 95% of such intervals containing the true mean if the process is repeated.

Step-by-step explanation:

To calculate a 95% confidence interval for the mean number of minutes students concentrate on their professor during a lecture, we use a t-distribution because the sample size is smaller than 30, or the population standard deviation is unknown. In this case, we have a sample of 105 students, and the sample mean is 36.2 minutes with a standard deviation of 13.1 minutes.

The formula for the confidence interval is:

Confidence Interval = Sample Mean ± (t-value * Standard Deviation / √n)

The degrees of freedom (df) would be 104 (n-1). The appropriate t-value for a two-tailed test at a 95% confidence interval can be found using a t-distribution table or statistical software.

Once the t-value is known, we plug in the numbers:

Confidence Interval = 36.2 ± (t-value * 13.1 / √105)

To answer part b, the confidence interval would provide the range in which we are 95% confident the true mean concentration time lies. Part c suggests if we repeated this study with many groups of 105 students, we would get different confidence intervals. For part d, about 95% of these intervals will contain the true population mean, and about 5% will not.