A researcher is interested in finding a 95\% confidence interval for the mean number of times per day that college students text. The study included 131 students who averaged 25.5 texts per day. The standard deviation was 22.1 texts.
a. To compute the confidence interval use a $t$ distribution.
b. With $95 \%$ confidence the population mean number of texts per day is between and texts.
c. If many groups of 131 randomly selected members are studied, then a different confidence interval would be produced from each group. About 95 percent of these confidence intervals will contain the true population number of texts per day and about 5 percent will not contain the true population mean number of texts per day.
Enter an integer or decimal number, accurate to at least 3 decimal places [more..)
Answer
\(\boxed{\text{Final Answer: With 95% confidence, the population mean number of texts per day is between 21.680 and 29.320.}}\)
Step 1 ：Given that the sample mean (\(\bar{x}\)) is 25.5, the sample standard deviation (s) is 22.1, and the sample size (n) is 131, we want to find a 95% confidence interval for the mean number of times per day that college students text.
Step 2 ：We use the formula for a confidence interval for a population mean with a known standard deviation: \(\bar{x} \pm t_{\frac{\alpha}{2}, n-1} \cdot \frac{s}{\sqrt{n}}\)
Step 3 ：In this formula, \(t_{\frac{\alpha}{2}, n-1}\) is the t-score for a two-tailed test with significance level \(\alpha\) and \(n-1\) degrees of freedom. Since we want a 95% confidence interval, \(\alpha = 0.05\), and we need to find the value of \(t_{0.025, 130}\).
Step 4 ：Using a t-distribution table or a statistical software, we find that \(t_{0.025, 130} = 1.9783804054271528\).
Step 5 ：Substituting the given values and the calculated t-score into the formula, we get the confidence interval: \((25.5 - 1.9783804054271528 \cdot \frac{22.1}{\sqrt{131}}, 25.5 + 1.9783804054271528 \cdot \frac{22.1}{\sqrt{131}})\)
Step 6 ：Solving the above expression, we get the confidence interval: \((21.67997304129269, 29.32002695870731)\)
Step 7 ：\(\boxed{\text{Final Answer: With 95% confidence, the population mean number of texts per day is between 21.680 and 29.320.}}\)
