You are testing the claim that the mean GPA of night students is less than the mean GPA of day students.

You sample 45 night students, and the sample mean GPA is 2.18 with a standard deviation of 0.31

You sample 30 day students, and the sample mean GPA is 1.97 with a standard deviation of 0.71

Calculate the test statistic, rounded to 2 decimal places

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√n

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability
of the Z score.

6.78

To compare the means of the two groups, we can use a t-test.

The formula for the t-test statistic is:

t = (x1 - x2) / sqrt((s1^2 / n1) + (s2^2 / n2))

Where:
x1 and x2 are the sample means,
s1 and s2 are the sample standard deviations,
n1 and n2 are the sample sizes.

Given:
x1 = 2.18 (mean GPA of night students)
s1 = 0.31 (standard deviation of night students)
n1 = 45 (sample size of night students)

x2 = 1.97 (mean GPA of day students)
s2 = 0.71 (standard deviation of day students)
n2 = 30 (sample size of day students)

Substituting the given values into the formula, we have:

t = (2.18 - 1.97) / sqrt((0.31^2 / 45) + (0.71^2 / 30))

Calculating this expression yields the test statistic:
t ≈ 0.21 / sqrt(0.001 / 45 + 0.005 / 30)

Rounding to 2 decimal places:
t ≈ 0.21 / sqrt(0.00002222 + 0.00016667)
t ≈ 0.21 / sqrt(0.00018889)
t ≈ 0.21 / 0.013734
t ≈ 15.31

Therefore, the test statistic, rounded to 2 decimal places, is approximately 15.31.

To test the claim that the mean GPA of night students is less than the mean GPA of day students, we can use a t-test to compare the two sample means.

The formula for calculating the t-test statistic is:
t = (x1 - x2) / sqrt((s1^2/n1) + (s2^2/n2))

Where:
x1 = sample mean of night students
x2 = sample mean of day students
s1 = standard deviation of night students
s2 = standard deviation of day students
n1 = sample size of night students
n2 = sample size of day students

Given the following data:

x1 = 2.18 (sample mean of night students)
x2 = 1.97 (sample mean of day students)
s1 = 0.31 (standard deviation of night students)
s2 = 0.71 (standard deviation of day students)
n1 = 45 (sample size of night students)
n2 = 30 (sample size of day students)

Let's substitute these values into the formula:

t = (2.18 - 1.97) / sqrt((0.31^2/45) + (0.71^2/30))

Calculating the values inside the square root:

t = (0.21) / sqrt((0.0097) + (0.016))

Simplifying further:

t = 0.21 / sqrt(0.0257)

t = 0.21 / 0.1603

t ≈ 1.31 (rounded to 2 decimal places)

Therefore, the test statistic for this claim is approximately 1.31.