QuestionAssume that adults were randomly selected for a poll. They were asked if they "favor or oppose using federal tax dollars to fund medical research using stem cells obtained from human embryos." Of those polled, 489 were in favor, 398 were opposed, and 122 were unsure. A politician claims that people don't really understand the stem cell issue and their responses to such questions are random responses equivalent to a coin toss. Exclude the 122 subjects who said that they were unsure, and use a 0.10 significance level to test the claim that the proportion of subjects who respond in favor is equal to 0.5 . What does the result suggest about the politician's claim?
The test statistic for this hypothesis test is 3.04
(Round to two decimal places as needed.)
Identify the P -value for this hypothesis test.
The P-value for this hypothesis test is 0.002 .
(Round to three decimal places as needed.)
Identify the conclusion for this hyporitesis test.
A. Fail to reject . There is not sufficient evidence to warrant rejection of the claim that the responses are equivalent to a coin toss.
B. Fail to reject . There is sufficient evidence to warrant rejection of the claim that the responses are equivalent to a coin toss.
C. Reject . There is not sufficient evidence to warrant rejection of the claim that the responses are equivalent to a coin toss.
D. Reject . There is sufficient evidence to warrant rejection of the claim that the responses are equivalent to a coin toss.
Studdy Solution
STEP 1
What is this asking?
Is the politician right to think that people's opinions on using federal tax dollars for embryonic stem cell research are just random, like flipping a coin, based on a poll (ignoring the "unsure" folks)?
Watch out!
Don't mix up the *number* of people who responded a certain way with the *proportion* of people who responded that way.
Also, remember that a small p-value means strong evidence *against* the politician's claim!
STEP 2
1. Set up the hypothesis test
2. Calculate the sample proportion
3. Interpret the p-value and draw a conclusion
STEP 3
Our **null hypothesis** () is what we're trying to disprove.
In this case, the politician thinks the responses are random, like a coin toss.
So, if the responses are random, we'd expect half the people to be in favor.
That means our null hypothesis is , where is the **true proportion** of people in favor.
STEP 4
The **alternative hypothesis** () is what we suspect might be true if the null hypothesis is wrong.
Here, it's simply that the true proportion isn't 0.5.
So, .
This is a **two-tailed test** because we're looking for any difference, whether the proportion is higher or lower than 0.5.
STEP 5
We're told to exclude the **122 unsure** people.
So, we have **489 in favor** and **398 opposed**, for a total of responses.
STEP 6
The **sample proportion** () is the number of "successes" (people in favor) divided by the total number of relevant responses.
So, .
This is the proportion of people in our sample who are in favor.
STEP 7
The problem tells us the **p-value is 0.002**.
Remember, the p-value is the probability of getting results as extreme as ours (or even more extreme) *if* the null hypothesis were true.
A tiny p-value means it's super unlikely we'd see these results if the politician were right.
STEP 8
Our **significance level** is .
Since our **p-value (0.002)** is *much* smaller than our **significance level (0.10)**, we have strong evidence to **reject the null hypothesis**.
STEP 9
Because we **reject** , we conclude that there *is* sufficient evidence to warrant rejection of the claim that the responses are equivalent to a coin toss.
It looks like people aren't just guessing randomly!
STEP 10
The p-value is **0.002**.
We **reject** the null hypothesis.
The correct answer is **D**.
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