Question

at a local community college, the mean gpa of daytime students is 3 with a standard deviation of 0.3. the administrators believe the night students have a lower gpa than the day students, so they collect a random sample of 56 night students and calculated their mean gpa to be 2.96. perform a hypothesis test at the 5% significance level.
what are the null and alternative hypotheses?
$h_0: mu = square$
$h_a: mu square$
what type of hypothesis test is this?
\bigcirc two - tailed
\bigcirc right - tailed
\bigcirc left - tailed
what is the value of the test statistic? $z=square$ (round to 2 decimal places)

Explanation:

Part 1: Null and Alternative Hypotheses

Step 1: Define Null Hypothesis

The null hypothesis \( H_0 \) is a statement of no difference or equality. Here, the mean GPA of daytime students is 3, so we assume the mean GPA of night students is equal to 3 in the null hypothesis. So \( H_0: \mu = 3 \).

Step 2: Define Alternative Hypothesis

The administrators believe night students have a lower GPA than day students. So the alternative hypothesis \( H_a \) is that the mean GPA of night students \( \mu \) is less than 3, i.e., \( H_a: \mu < 3 \).

Step 1: Analyze the Alternative Hypothesis

The alternative hypothesis \( H_a: \mu < 3 \) indicates that we are testing if the mean is less than a certain value. This corresponds to a left - tailed test because we are interested in the area in the left - tail of the distribution (where values are less than the hypothesized mean).

Step 1: Recall the z - test formula for the mean

The formula for the z - test statistic when the population standard deviation \( \sigma \) is known is \( z=\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}} \), where \( \bar{x} \) is the sample mean, \( \mu \) is the population mean under the null hypothesis, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

Step 2: Identify the values

We know that \( \bar{x} = 2.96 \), \( \mu = 3 \), \( \sigma=0.3 \), and \( n = 56 \).

Step 3: Substitute the values into the formula

First, calculate the denominator \( \frac{\sigma}{\sqrt{n}}=\frac{0.3}{\sqrt{56}}\approx\frac{0.3}{7.4833}\approx0.0401 \)
Then, calculate the numerator \( \bar{x}-\mu=2.96 - 3=- 0.04 \)
Now, calculate the z - score: \( z=\frac{-0.04}{0.0401}\approx - 0.9975\approx - 1.00 \) (rounded to two decimal places)

Answer:

\( H_0: \mu = 3 \)
\( H_a: \mu < 3 \)

Part 2: Type of Hypothesis Test