Math / Data & Statistics

Question4. It has been estimated that the G-car obtains a mean of 35 miles per gallon on the highway, and the company that manufactures the car claims that it exceeds this estimate in highway driving. To support its assertion, the company randomly selects 36 G -cars and records the mileage obtained for each car over a driving course similar to that used to obtain the estimate. The following data resulted: $\bar{x}=36.8$ miles per gallon, $s=6$ miles per gallon. Calculate the value of $\beta$ if the true value of the mean is 37 miles per gallon. Use $\alpha=0.025$ .

Studdy Solution

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.

STEP 11

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

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Studdy Solution

STEP 1

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

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.

STEP 11

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

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Studdy Solution

STEP 1

STEP 2

1. Set up the hypothesis test2. Find the critical value3. Calculate the non-rejection range4. Calculate β\betaβ

STEP 3

We're testing if the car's mileage is *greater than* 35 mpg.So, our **null hypothesis** H0H_0H0​ is μ≤35\mu \le 35μ≤35, and our **alternative hypothesis** HaH_aHa​ is μ>35\mu > 35μ>35.This is a **one-tailed test**!

STEP 4

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

STEP 5

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

STEP 6

With α=0.025\alpha = 0.025α=0.025 and df=35df = 35df=35, our **critical** \textit{t}-**value** is approximately t0.025,35≈2.03t_{0.025, 35} \approx 2.03t0.025,35​≈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

STEP 8

We *fail to reject* H0H_0H0​ if our sample mean xˉ\bar{x}xˉ is *less than or equal to* **37.03**.

STEP 9

STEP 10

STEP 11

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

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

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!

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

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

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.

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

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