Question

test the claim that the average gpa of rcc night students is smaller than 2.3 at the 0.05 significance level.
pick the correct pair of hypotheses (null and alternative hypothesis) from the choices below.
$h_0: p \leq 0.575$ $h_0: p \geq 0.575$ $h_0: p = 0.575$ $h_0: \mu \leq 2.3$ $h_0: \mu = 2.3$ $h_0: \mu \geq 2.3$
$h_1: p > 0.575$ $h_1: p < 0.575$ $h_1: p \
eq 0.575$ $h_1: \mu > 2.3$ $h_1: \mu \
eq 2.3$ $h_1: \mu < 2.3$
based on a sample of 40 people, the samples average gpa was 2.26 with a standard deviation of 0.02.
the test statistic is: this is not a decimal or integer value. (to 2 decimals)
the p value is: 0.00 (to 2 decimals)
based on this we:

• reject the null hypothesis
• fail to reject the null hypothesis

what is a type 1 error in this context?

• if you conclude that the average gpa is not smaller 2.3 when it really is.
• if you conclude that the average gpa is smaller 2.3 when it really is not

Explanation:

Step1: Identify Hypotheses

The claim is about the average GPA (\(\mu\)) being smaller than 2.3. So, the null hypothesis (\(H_0\)) should be the opposite or equality, and the alternative (\(H_1\)) is the claim. Thus, \(H_0: \mu \geq 2.3\) and \(H_1: \mu < 2.3\).

Step2: Calculate Test Statistic

We use the t - test formula (since population standard deviation is unknown, use sample std dev \(s = 0.02\), sample mean \(\bar{x}=2.26\), population mean under \(H_0\) \(\mu_0 = 2.3\), sample size \(n = 40\)):

$$ t=\frac{\bar{x}-\mu_0}{\frac{s}{\sqrt{n}}} $$

Substitute values: \(\bar{x}=2.26\), \(\mu_0 = 2.3\), \(s = 0.02\), \(n = 40\)

$$ t=\frac{2.26 - 2.3}{\frac{0.02}{\sqrt{40}}}=\frac{- 0.04}{\frac{0.02}{6.3246}}\approx\frac{-0.04}{0.00316}\approx - 12.66 $$

(Note: The negative sign indicates direction, but magnitude is what's key for test statistic here. Rounded to 2 decimals, it's - 12.66, but if we consider absolute value or just the calculated value, the test statistic is approximately - 12.66)

Step3: Analyze p - value and Decision

The p - value is 0.00 (to 2 decimals), which is less than \(\alpha = 0.05\), so we reject \(H_0\).

Step4: Type I Error

Type I error is rejecting \(H_0\) when it's true. So, concluding average GPA is smaller than 2.3 when it's not.

Answer:

s:

1. Correct Hypotheses: \(H_0: \mu \geq 2.3\), \(H_1: \mu < 2.3\)
2. Test Statistic: \(\boldsymbol{-12.66}\) (or 12.66 in absolute value context)
3. p - value: \(0.00\)
4. Decision: Reject the null hypothesis
5. Type I Error: Concluding average GPA is smaller than 2.3 when it really is not