Question

13 multiple choice 1 point a random sample of 100 customers at a local bank showed that their average service time was 4 minutes with a sample standard deviation of 1.3 minutes. find the 95% confidence interval for the true average service time. 4 ± 0.2139 1.3 ± 0.2548 4 ± 0.2548 4 ± 1.3

Explanation:

Step1: Identify the formula

For a large - sample (n ≥ 30) confidence interval of the population mean when the population standard - deviation is unknown (we use the sample standard deviation s instead), the formula is $\bar{x}\pm z_{\alpha/2}\frac{s}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $z_{\alpha/2}$ is the z - value, s is the sample standard deviation, and n is the sample size.

Step2: Determine the values

We are given that $\bar{x} = 4$ (sample mean), s = 1.3 (sample standard deviation), and n = 100 (sample size). For a 95% confidence interval, $\alpha=1 - 0.95 = 0.05$, so $\alpha/2=0.025$. The $z$ - value $z_{\alpha/2}=z_{0.025}=1.96$.

Step3: Calculate the margin of error

The margin of error E is $z_{\alpha/2}\frac{s}{\sqrt{n}}$. Substitute the values: $1.96\times\frac{1.3}{\sqrt{100}}=1.96\times\frac{1.3}{10}=1.96\times0.13 = 0.2548$.

Step4: Write the confidence interval

The 95% confidence interval is $\bar{x}\pm E=4\pm0.2548$.

Answer:

$4\pm0.2548$