Question

a random sample of 22 college mens basketball games during the last season had an average attendance of 5,132 with a sample standard deviation of 1,756. complete parts a and b below.
a. construct a 90% confidence interval to estimate the average attendance of a college mens basketball game during the last season.
the 90% confidence interval to estimate the average attendance of a college mens basketball game during the last season is from a lower limit of 4,488 to an upper limit of 5,776.
(round to the nearest whole numbers.)
b. what assumptions need to be made about this population?
a. the only assumption needed is that the population follows the students t - distribution.
b. the only assumption needed is that the population size is larger than 30.
c. the only assumption needed is that the population follows the normal distribution.
d. the only assumption needed is that the population distribution is skewed to one side.

Explanation:

Step1: Identify the formula

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

Step2: Determine the values

We have $n = 22$, $\bar{x}=5132$, $s = 1756$. The degrees of freedom $df=n - 1=22-1 = 21$. For a 90% confidence interval, $\alpha=1 - 0.90 = 0.10$ and $\alpha/2=0.05$. Looking up in the t - distribution table, $t_{0.05,21}=1.721$.

Step3: Calculate the margin of error

The margin of error $E=t_{\alpha/2}\frac{s}{\sqrt{n}}=1.721\times\frac{1756}{\sqrt{22}}\approx1.721\times\frac{1756}{4.69}=1.721\times374.41\approx644$.

Step4: Calculate the confidence interval

The lower limit is $\bar{x}-E=5132 - 644=4488$ and the upper limit is $\bar{x}+E=5132 + 644=5776$.

For part b:
When constructing a confidence interval for the population mean using the t - distribution (since the population standard deviation is unknown and we have a small sample size $n<30$), we assume that the population from which the sample is drawn follows a normal distribution.

Answer:

a. The 90% confidence interval is from 4488 to 5776.
b. C. The only assumption needed is that the population follows the normal distribution.