Question

the grade point average collected from a random sample of 150 students producing a mean of 2.86. assume that the population standard deviation is 0.78. construct the confidence interval for the population mean, μ, if c = 0.98.
a. (2.71,3.01)
b. (2.31,3.88)
c. (2.43,3.79)
d. (2.51,3.53)

Explanation:

Step1: Find the significance level $\alpha$

Given confidence level $c = 0.98$.
$\alpha=1 - c=1 - 0.98 = 0.02$.

Step2: Find $z_{\alpha/2}$

$\alpha/2=\frac{0.02}{2}=0.01$.
Looking up in the standard normal distribution table, $z_{0.01}\approx2.33$.

Step3: Calculate the margin of error $E$

Population standard deviation $\sigma = 0.78$, sample size $n = 150$.
$E=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=2.33\times\frac{0.78}{\sqrt{150}}$.
First, $\sqrt{150}\approx12.25$.
$\frac{0.78}{12.25}\approx0.0637$.
$E = 2.33\times0.0637\approx0.15$.

Step4: Calculate the confidence interval

Sample mean $\bar{x}=2.86$.
The confidence interval is $\bar{x}-E<\mu<\bar{x} + E$.
$2.86-0.15<\mu<2.86 + 0.15$.
$2.71<\mu<3.01$.

Answer:

A. $(2.71,3.01)$