Math / Data & Statistics

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? A. $\mathrm{H}_{0}: p=0.5$ $H_{1}: p<0.5$ B. $H_{0}: p=0.5$ $H_{1}: p \neq 0.5$ C. $\begin{array}{l} H_{0}: p=0.5 \\ H_{1}: p>0.5 \end{array}$ D. $\begin{array}{l} H_{0}: p \neq 0.5 \\ H_{1}: p=0.5 \end{array}$
Identify the test statistic for this hypothesis test. 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 $\square$ (Round to three decimal places as needed.)

Studdy Solution

STEP 1

What is this asking? A politician thinks people's opinions on using federal taxes for stem cell research are just random, like flipping a coin.
We need to check if the actual opinions are really that random. Watch out! Don't forget we're only looking at the people who had an opinion (favor or oppose), not the unsure folks.
Also, the politician isn't saying people are more likely to be in favor or opposed, just that it's different from a 50/50 split, so it's a two-tailed test!

STEP 2

1. Set up the hypothesis test
2. Calculate the sample proportion
3. Calculate the test statistic
4. Calculate the P-value
5. Interpret the P-value

STEP 3

The null hypothesis ( $H_0$ ) is what we're trying to disprove.
In this case, the politician claims the proportion of people in favor is 0.5 (like a coin toss).
So, $H_0: p = 0.5$ .

STEP 4

The alternative hypothesis ( $H_1$ ) is what we suspect is true if the null hypothesis is false.
Here, we're testing if the proportion is different from 0.5, so $H_1: p \neq 0.5$ .
This makes it a two-tailed test.

STEP 5

We're excluding the unsure people, so we have 489 in favor and 398 opposed.
That's $489 + 398 = 887$ total responses.

STEP 6

The sample proportion ( $\hat{p}$ ) is the number in favor divided by the total number of responses: $\hat{p} = \frac{489}{887} \approx 0.551$ .

STEP 7

The test statistic (z) tells us how far our sample proportion is from the claimed proportion, in terms of standard deviations.
The formula is: $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ where $p_0$ is the proportion under the null hypothesis (0.5 in our case) and $n$ is the sample size (887).

STEP 8

$z = \frac{0.551 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{887}}} = \frac{0.051}{\sqrt{\frac{0.25}{887}}} \approx \frac{0.051}{0.0168} \approx 3.04$

STEP 9

The P-value is the probability of getting a sample proportion as extreme as ours (or more extreme) if the null hypothesis were true.
Since it's a two-tailed test, we need to consider both tails.

STEP 10

Our test statistic is 3.04.
Using a z-table or calculator, the area to the right of 3.04 is approximately 0.0012.

STEP 11

Since it's a two-tailed test, we double this area: $2 \cdot 0.0012 = 0.0024$ .

STEP 12

Our P-value (0.0024) is less than the significance level (0.10).

STEP 13

This means we reject the null hypothesis.
The evidence suggests the proportion of people in favor is not 0.5.

STEP 14

The test statistic is 3.04.
The P-value is 0.002.
The result suggests that the politician's claim is likely wrong.
People's opinions on stem cell research aren't just random!

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Studdy Solution

STEP 1

STEP 2

1. Set up the hypothesis test
2. Calculate the sample proportion
3. Calculate the test statistic
4. Calculate the P-value
5. Interpret the P-value

STEP 3

The null hypothesis ( $H_0$ ) is what we're trying to disprove.
In this case, the politician claims the proportion of people in favor is 0.5 (like a coin toss).
So, $H_0: p = 0.5$ .

STEP 4

The alternative hypothesis ( $H_1$ ) is what we suspect is true if the null hypothesis is false.
Here, we're testing if the proportion is different from 0.5, so $H_1: p \neq 0.5$ .
This makes it a two-tailed test.

STEP 5

We're excluding the unsure people, so we have 489 in favor and 398 opposed.
That's $489 + 398 = 887$ total responses.

STEP 6

The sample proportion ( $\hat{p}$ ) is the number in favor divided by the total number of responses: $\hat{p} = \frac{489}{887} \approx 0.551$ .

STEP 7

STEP 8

$z = \frac{0.551 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{887}}} = \frac{0.051}{\sqrt{\frac{0.25}{887}}} \approx \frac{0.051}{0.0168} \approx 3.04$

STEP 9

The P-value is the probability of getting a sample proportion as extreme as ours (or more extreme) if the null hypothesis were true.
Since it's a two-tailed test, we need to consider both tails.

STEP 10

Our test statistic is 3.04.
Using a z-table or calculator, the area to the right of 3.04 is approximately 0.0012.

STEP 11

Since it's a two-tailed test, we double this area: $2 \cdot 0.0012 = 0.0024$ .

STEP 12

Our P-value (0.0024) is less than the significance level (0.10).

STEP 13

This means we reject the null hypothesis.
The evidence suggests the proportion of people in favor is not 0.5.

STEP 14

The test statistic is 3.04.
The P-value is 0.002.
The result suggests that the politician's claim is likely wrong.
People's opinions on stem cell research aren't just random!

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Studdy Solution

STEP 1

STEP 2

1. Set up the hypothesis test2. Calculate the sample proportion3. Calculate the test statistic4. Calculate the P-value5. Interpret the P-value

STEP 3

The **null hypothesis** (H0H_0H0​) is what we're trying to disprove.In this case, the politician claims the proportion of people in favor is 0.5 (like a coin toss).So, H0:p=0.5H_0: p = 0.5H0​:p=0.5.

STEP 4

The **alternative hypothesis** (H1H_1H1​) is what we suspect is true if the null hypothesis is false.Here, we're testing if the proportion is *different* from 0.5, so H1:p≠0.5H_1: p \neq 0.5H1​:p=0.5.This makes it a **two-tailed test**.

STEP 5

We're excluding the unsure people, so we have **489** in favor and **398** opposed.That's 489+398=887489 + 398 = 887489+398=887 total responses.

STEP 6

The **sample proportion** (p^\hat{p}p^​) is the number in favor divided by the total number of responses: p^=489887≈0.551\hat{p} = \frac{489}{887} \approx 0.551p^​=887489​≈0.551.

STEP 7

STEP 8

z=0.551−0.50.5(1−0.5)887=0.0510.25887≈0.0510.0168≈3.04z = \frac{0.551 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{887}}} = \frac{0.051}{\sqrt{\frac{0.25}{887}}} \approx \frac{0.051}{0.0168} \approx 3.04z=8870.5(1−0.5)​​0.551−0.5​=8870.25​​0.051​≈0.01680.051​≈3.04

STEP 9

The **P-value** is the probability of getting a sample proportion as extreme as ours (or more extreme) *if* the null hypothesis were true.Since it's a two-tailed test, we need to consider both tails.

STEP 10

Our test statistic is **3.04**.Using a z-table or calculator, the area to the right of 3.04 is approximately **0.0012**.

STEP 11

Since it's a two-tailed test, we double this area: 2⋅0.0012=0.00242 \cdot 0.0012 = 0.00242⋅0.0012=0.0024.

STEP 12

Our **P-value** (0.0024) is less than the **significance level** (0.10).

STEP 13

This means we **reject the null hypothesis**.The evidence suggests the proportion of people in favor is *not* 0.5.

STEP 14

The test statistic is **3.04**.The P-value is **0.002**.The result suggests that the politician's claim is likely wrong.People's opinions on stem cell research aren't just random!

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1. Set up the hypothesis test
2. Calculate the sample proportion
3. Calculate the test statistic
4. Calculate the P-value
5. Interpret the P-value

The null hypothesis ( $H_0$ ) is what we're trying to disprove.
In this case, the politician claims the proportion of people in favor is 0.5 (like a coin toss).
So, $H_0: p = 0.5$ .

The alternative hypothesis ( $H_1$ ) is what we suspect is true if the null hypothesis is false.
Here, we're testing if the proportion is different from 0.5, so $H_1: p \neq 0.5$ .
This makes it a two-tailed test.

We're excluding the unsure people, so we have 489 in favor and 398 opposed.
That's $489 + 398 = 887$ total responses.

The sample proportion ( $\hat{p}$ ) is the number in favor divided by the total number of responses: $\hat{p} = \frac{489}{887} \approx 0.551$ .

$z = \frac{0.551 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{887}}} = \frac{0.051}{\sqrt{\frac{0.25}{887}}} \approx \frac{0.051}{0.0168} \approx 3.04$

The P-value is the probability of getting a sample proportion as extreme as ours (or more extreme) if the null hypothesis were true.
Since it's a two-tailed test, we need to consider both tails.

Our test statistic is 3.04.
Using a z-table or calculator, the area to the right of 3.04 is approximately 0.0012.

Since it's a two-tailed test, we double this area: $2 \cdot 0.0012 = 0.0024$ .

Our P-value (0.0024) is less than the significance level (0.10).

This means we reject the null hypothesis.
The evidence suggests the proportion of people in favor is not 0.5.

The test statistic is 3.04.
The P-value is 0.002.
The result suggests that the politician's claim is likely wrong.
People's opinions on stem cell research aren't just random!