Question

use the standard normal distribution or the t - distribution to construct a 99% confidence interval for the population mean. justify your decision. if neither distribution can be used, explain why. interpret the results.
in a recent season, the population standard deviation of the yards per carry for all running backs was 1.27. the yards per carry of 25 randomly selected running backs are shown below. assume the yards per carry are normally distributed.
1.3 2.9 5.3 6.6 6.5 3.1 5.9 7.4 5.6 1.6 1.4 5.9 5.3
7.1 2.8 2.7 5.9 6.8 5.7 4.2 5.6 4.1 5.9 3.1 6.9
which distribution should be used to construct the confidence interval?
a. use a t - distribution because n < 30 and σ is known.
b. use a t - distribution because n < 30 and σ is unknown.
c. use a normal distribution because σ is known and the data are normally distributed.
d. use a normal distribution because n < 30, the data are normally distributed and σ is unknown.
e. cannot use the standard normal distribution or the t - distribution because σ is unknown, n < 30, and the data are not normally distributed.
select the correct choice below and, if necessary, fill in any answer boxes to complete your choice.
a. the 99% confidence interval is (round to two decimal places as needed.)
b. neither distribution can be used to construct the confidence interval.

Explanation:

Step1: Calculate the sample mean

The formula for the sample mean $\bar{x}$ is $\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}$.
Sum of the data:

$$
LATEXBLOCK0
$$

Since $n = 25$, $\bar{x}=\frac{125.5}{25}=5.02$.

Step2: Find the critical value

For a 99% confidence interval and using the standard normal distribution (because $\sigma$ is known and data is normal), the critical value $z_{\alpha/2}$.
Since $\alpha=1 - 0.99 = 0.01$, then $\alpha/2=0.005$. Looking up in the standard - normal table, $z_{0.005}=2.576$.

Step3: Calculate the margin of error

The formula for the margin of error $E$ is $E = z_{\alpha/2}\frac{\sigma}{\sqrt{n}}$.
Given $\sigma = 1.27$ and $n = 25$, $E=2.576\times\frac{1.27}{\sqrt{25}}=2.576\times\frac{1.27}{5}=2.576\times0.254 = 0.654304$.

Step4: Calculate the confidence interval

The formula for the confidence interval is $\bar{x}-E\lt\mu\lt\bar{x}+E$.
Substitute $\bar{x}=5.02$ and $E = 0.654304$:
$5.02-0.654304\lt\mu\lt5.02 + 0.654304$.
$4.37\lt\mu\lt5.67$ (rounded to two decimal places).

Answer:

A. The 99% confidence interval is $(4.37,5.67)$