Math  /  Data & Statistics

Question26) Formulate the indicated conclusion in nontechnical terms. Be sure to address the original claim. Carter Motor Company claims that its new sedan, the Libra, will average better than 32 miles 26) \qquad per gallon in the city. Assuming that a hypothesis test of the claim has been conducted and that the conclusion is to reject the null hypothesis, state the conclusion in nontechnical terms. A) There is sufficient evidence to support the claim that the mean is less than 32 miles per gallon. B) There is not sufficient evidence to support the claim that the mean is less than 32 miles per gallon. C) There is sufficient evidence to support the claim that the mean is greater than 32 miles per gallon. D) There is not sufficient evidence to support the claim that the mean is greater than 32 miles pergallon. 27) Use the given information to find the PP-value. Also, use a 0.05 significance level and state the conclusion about the null hypothesis (reject the null hypothesis or fail to reject the null 27) \qquad hypothesis). With H1:p3/5H_{1}: p \neq 3 / 5, the test statistic is z=0.78z=0.78. A) 0.2177 ; reject the null hypothesis B) 0.4354 ; reject the null hypothesis

Studdy Solution

STEP 1

What is this asking? We need to translate some statistical jargon into plain English and also find a p-value and determine what it means for a hypothesis. Watch out! Don't mix up "reject the null hypothesis" and "accept the alternative hypothesis".
Also, remember that p-values measure the probability of observing our data *if* the null hypothesis were true.

STEP 2

1. Translate the Conclusion
2. Calculate the P-value
3. Interpret the P-value

STEP 3

Carter Motor Company believes their new car, the Libra, gets better than 3232 miles per gallon in the city.
That's their claim!

STEP 4

The **null hypothesis** is usually the opposite of what we're trying to prove.
Here, it would be that the Libra's average gas mileage is *not* better than 3232 mpg.
The **alternative hypothesis** is what we're trying to prove: that the Libra *does* get better than 3232 mpg.

STEP 5

We're told to assume the **null hypothesis** is rejected.
This means there's enough evidence to suggest the null hypothesis is wrong.
In simpler terms, the data suggests the Libra *does* get better than 3232 mpg.

STEP 6

The p-value tells us how likely it is to see the data we collected (or even more extreme data) *if* the null hypothesis were actually true.
A small p-value means it's unlikely, which makes us doubt the null hypothesis.

STEP 7

Our alternative hypothesis, H1:p3/5H_1: p \neq 3/5, is a **two-tailed test**.
This means we're looking for evidence that the true value is *different* from 3/53/5, either greater than or less than.

STEP 8

We're given a test statistic z=0.78z = 0.78.
We need to find the area to the *right* of z=0.78z = 0.78 under the standard normal curve.
Using a z-table or calculator, we find this area to be approximately 10.7823=0.21771 - 0.7823 = 0.2177.

STEP 9

Since it's a two-tailed test, we multiply the one-tail area by 22 to get the p-value: 20.2177=0.43542 \cdot 0.2177 = 0.4354.
So, our **p-value** is **0.43540.4354**.

STEP 10

We're given a significance level of 0.050.05.
This is our threshold for deciding whether to reject the null hypothesis.

STEP 11

Our p-value (0.43540.4354) is *greater* than the significance level (0.050.05).

STEP 12

Since the p-value is greater than the significance level, we *fail to reject* the null hypothesis.
This means we don't have enough evidence to say the true proportion is different from 3/53/5.

STEP 13

For problem 26, the correct answer is C: There is sufficient evidence to support the claim that the mean is greater than 32 miles per gallon.
For problem 27, the p-value is 0.43540.4354.
Since 0.4354>0.050.4354 > 0.05, we fail to reject the null hypothesis.

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