Final answer:
The question involves calculating a 95% confidence interval for a sample mean using given statistics, including the sample size (n), sample standard deviation (s), and degrees of freedom (df).
The confidence interval formula incorporates the critical value from a t-distribution and requires using statistical software or calculators for computation. Sample size variations will affect the width of the confidence interval.
Step-by-step explanation:
The student's question relates to constructing a 95% confidence interval for the mean of a sample size of n = 10, with a given sample mean (not specified in the question, but represented by the symbol for a sample mean) and a sample standard deviation (s) of 3.7. To construct a confidence interval, we use the formula for the confidence interval which is the sample mean ± the margin of error.
The margin of error is calculated using the critical value from the t-distribution (since the population standard deviation is unknown) multiplied by the sample standard deviation divided by the square root of the sample size.
determine the degrees of freedom as df = n - 1. Secondly, we need the critical t-value for the 95% confidence level, which we can find in a t-distribution table or by using statistical software or calculators. Inserting the mean, the critical t-value, the sample standard deviation, and the sample size into the formula gives us the actual confidence interval.
The process outlined for calculating a confidence interval involves selecting the appropriate statistical tests on calculators or software, inputting the necessary statistics (sample standard deviation, sample mean, sample size, and confidence level), and then choosing the calculate option to get the confidence interval.
As the sample size changes, the width of the confidence interval will adjust; with a larger sample size, the interval tends to get narrower (indicating more precision), and with a smaller sample size, the interval gets wider (indicating less precision).