Math Snap

STEP 1

What is this asking?
We want to find the probability of not rejecting a false null hypothesis, specifically the chance we'd wrongly conclude that the G-car doesn't get better than 35 mpg when it actually gets 37 mpg.
Watch out!
Don't mix up $\beta$ (the probability of a Type II error) with $\alpha$ (the probability of a Type I error, also known as the significance level)!

STEP 2

1. Set up the hypothesis test
2. Find the critical value
3. Calculate the non-rejection range
4. Calculate $\beta$

STEP 3

We're testing if the car's mileage is greater than 35 mpg.
So, our null hypothesis $H_0$ is $\mu \le 35$, and our alternative hypothesis $H_a$ is $\mu > 35$.
This is a one-tailed test!

STEP 4

Since we know the sample standard deviation ($s = 6$), we'll use a t-test.

STEP 5

Our sample size is $n = 36$, so our degrees of freedom are $df = n - 1 = 36 - 1 = 35$.

STEP 6

With $\alpha = 0.025$ and $df = 35$, our critical \textit{t}-value is approximately $t_{0.025, 35} \approx 2.03$.
We're looking for a value in the t-distribution table that leaves 0.025 of the area in the upper tail.

STEP 7

We only reject $H_0$ if our test statistic is greater than the critical value.
The non-rejection limit is where the test statistic equals the critical value.
Let's call this limit $\bar{x}_{limit}$.
We have $t = \frac{\bar{x}_{limit} - \mu_0}{s / \sqrt{n}}$, so $2.03 = \frac{\bar{x}_{limit} - 35}{6 / \sqrt{36}}$.
Solving for $\bar{x}_{limit}$, we get $\bar{x}_{limit} = 35 + 2.03 \cdot \frac{6}{6} = 35 + 2.03 = 37.03$.

STEP 8

We fail to reject $H_0$ if our sample mean $\bar{x}$ is less than or equal to 37.03.

STEP 9

We're told the true mean is $\mu = 37$.
Let's standardize this value assuming the null hypothesis is true ($\mu_0 = 35$) and using the non-rejection limit we just calculated: $t = \frac{37.03 - 37}{6/\sqrt{36}} = \frac{0.03}{1} = 0.03$.

STEP 10

We want to find the probability of not rejecting $H_0$ when the true mean is 37.
This is the probability that our test statistic is less than 0.03.
Looking this up in a t-distribution table with 35 degrees of freedom, we find $P(t < 0.03) \approx 0.51$.
So, $\beta \approx$ 0.51.

The probability of making a Type II error ($\beta$) when the true mean is 37 mpg is approximately 0.51.