Math / Data & Statistics

QuestionA television show conducted an experiment to study what happens when buttered toast is dropped on the floor. When 46 buttered slices of toast were dropped, 25 of them landed with the buttered side up and 21 landed with the buttered side down. Use a 0.10 significance level to test the claim that toast will land with the buttered side down $50 \%$ of the time. Use the P-value method. Use the normal distribution as an approximation to the binomial distribution. After that, supposing the intent of the experiment was to assess the claim that toast will land with the buttered side down more than $50 \%$ of the time, write a conclusion that addresses the intent of the experiment.
Let p denote the population proportion of all buttered toast that will land with the buttered side down when dropped. Identify the null and alternative hypotheses to test the claim that buttered toast will land with the buttered side down $50 \%$ of the time. $\mathrm{H}_{0}$ : p $\square$ $\square$ $\mathrm{H}_{1} \mathrm{p}$ $\square$ $\square$ (Type integers or decimals. Do not round.)

Studdy Solution

STEP 1

What is this asking? We want to see if dropping toast really leads to the buttered side landing down 50% of the time, and then check if it lands buttered side down more than 50% of the time. Watch out! Don't mix up the two different questions – one is about exactly 50%, the other is about more than 50%!
Also, remember that we're using a normal distribution to approximate the binomial distribution here.

STEP 2

1. Set up the hypotheses
2. Calculate $\hat{p}$
3. Calculate the test statistic
4. Calculate the P-value
5. Draw a conclusion for the first question
6. Draw a conclusion for the second question

STEP 3

Our null hypothesis $H_0$ is that the toast lands buttered side down 50% of the time.
So, $H_0: p = 0.5$ .

STEP 4

For the first question, we're testing if it's different from 50%, so our alternative hypothesis $H_1$ is $H_1: p \neq 0.5$ .
This is a two-tailed test.

STEP 5

We have 21 out of 46 toasts landing buttered side down.
So, our sample proportion $\hat{p}$ is $\hat{p} = \frac{21}{46} \approx 0.4565$ .

STEP 6

The test statistic is given by $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ , where $p_0$ is the value of $p$ under the null hypothesis (0.5), and $n$ is the sample size (46).

STEP 7

Plugging in the values, we get $z = \frac{0.4565 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{46}}} = \frac{-0.0435}{\sqrt{\frac{0.25}{46}}} \approx \frac{-0.0435}{0.0736} \approx -0.591$ .

STEP 8

Since this is a two-tailed test, we need to find the area in both tails.
First, let's find the area to the left of $z = -0.591$ .
Using a z-table or calculator, we find this area to be approximately 0.277.

STEP 9

Since it's a two-tailed test, we multiply this value by 2 to get the P-value: $P \approx 2 \cdot 0.277 = 0.554$ .

STEP 10

Our P-value (0.554) is greater than our significance level (0.10).

STEP 11

Therefore, we fail to reject the null hypothesis.
There's not enough evidence to say that the toast doesn't land buttered side down 50% of the time.

STEP 12

For the second question, the alternative hypothesis would be $H_1: p > 0.5$ , making it a right-tailed test.

STEP 13

The P-value for a one-tailed test would be approximately $1 - 0.277 = 0.723$ .
This is still greater than 0.10.

STEP 14

So, we again fail to reject the null hypothesis.
There's not enough evidence to conclude that the toast lands buttered side down more than 50% of the time.

STEP 15

For the first question (is it different from 50%?), we fail to reject the null hypothesis.
For the second question (is it more than 50%?), we also fail to reject the null hypothesis.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Set up the hypotheses
2. Calculate $\hat{p}$
3. Calculate the test statistic
4. Calculate the P-value
5. Draw a conclusion for the first question
6. Draw a conclusion for the second question

STEP 3

Our null hypothesis $H_0$ is that the toast lands buttered side down 50% of the time.
So, $H_0: p = 0.5$ .

STEP 4

For the first question, we're testing if it's different from 50%, so our alternative hypothesis $H_1$ is $H_1: p \neq 0.5$ .
This is a two-tailed test.

STEP 5

We have 21 out of 46 toasts landing buttered side down.
So, our sample proportion $\hat{p}$ is $\hat{p} = \frac{21}{46} \approx 0.4565$ .

STEP 6

The test statistic is given by $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ , where $p_0$ is the value of $p$ under the null hypothesis (0.5), and $n$ is the sample size (46).

STEP 7

Plugging in the values, we get $z = \frac{0.4565 - 0.5}{\sqrt{\frac{0.5(1-0.5)}{46}}} = \frac{-0.0435}{\sqrt{\frac{0.25}{46}}} \approx \frac{-0.0435}{0.0736} \approx -0.591$ .

STEP 8

Since this is a two-tailed test, we need to find the area in both tails.
First, let's find the area to the left of $z = -0.591$ .
Using a z-table or calculator, we find this area to be approximately 0.277.

STEP 9

Since it's a two-tailed test, we multiply this value by 2 to get the P-value: $P \approx 2 \cdot 0.277 = 0.554$ .

STEP 10

Our P-value (0.554) is greater than our significance level (0.10).

STEP 11

Therefore, we fail to reject the null hypothesis.
There's not enough evidence to say that the toast doesn't land buttered side down 50% of the time.

STEP 12

For the second question, the alternative hypothesis would be $H_1: p > 0.5$ , making it a right-tailed test.

STEP 13

The P-value for a one-tailed test would be approximately $1 - 0.277 = 0.723$ .
This is still greater than 0.10.

STEP 14

So, we again fail to reject the null hypothesis.
There's not enough evidence to conclude that the toast lands buttered side down more than 50% of the time.

STEP 15

For the first question (is it different from 50%?), we fail to reject the null hypothesis.
For the second question (is it more than 50%?), we also fail to reject the null hypothesis.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Set up the hypotheses2. Calculate p^\hat{p}p^​3. Calculate the test statistic4. Calculate the P-value5. Draw a conclusion for the first question6. Draw a conclusion for the second question

STEP 3

Our **null hypothesis** H0H_0H0​ is that the toast lands buttered side down 50% of the time.So, H0:p=0.5H_0: p = 0.5H0​:p=0.5.

STEP 4

For the first question, we're testing if it's *different* from 50%, so our **alternative hypothesis** H1H_1H1​ is H1:p≠0.5H_1: p \neq 0.5H1​:p=0.5.This is a **two-tailed test**.

STEP 5

We have 21 out of 46 toasts landing buttered side down.So, our **sample proportion** p^\hat{p}p^​ is p^=2146≈0.4565\hat{p} = \frac{21}{46} \approx 0.4565p^​=4621​≈0.4565.

STEP 6

The **test statistic** is given by z=p^−p0p0(1−p0)nz = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}z=np0​(1−p0​)​​p^​−p0​​, where p0p_0p0​ is the value of ppp under the null hypothesis (0.5), and nnn is the sample size (46).

STEP 7

STEP 8

Since this is a two-tailed test, we need to find the area in both tails.First, let's find the area to the left of z=−0.591z = -0.591z=−0.591.Using a z-table or calculator, we find this area to be approximately 0.277.

STEP 9

Since it's a two-tailed test, we multiply this value by 2 to get the **P-value**: P≈2⋅0.277=0.554P \approx 2 \cdot 0.277 = 0.554P≈2⋅0.277=0.554.

STEP 10

Our **P-value** (0.554) is greater than our **significance level** (0.10).

STEP 11

Therefore, we **fail to reject the null hypothesis**.There's not enough evidence to say that the toast *doesn't* land buttered side down 50% of the time.

STEP 12

For the second question, the **alternative hypothesis** would be H1:p>0.5H_1: p > 0.5H1​:p>0.5, making it a **right-tailed test**.

STEP 13

The **P-value** for a one-tailed test would be approximately 1−0.277=0.7231 - 0.277 = 0.7231−0.277=0.723.This is still greater than 0.10.

STEP 14

So, we again **fail to reject the null hypothesis**.There's not enough evidence to conclude that the toast lands buttered side down *more* than 50% of the time.

STEP 15

For the first question (is it different from 50%?), we fail to reject the null hypothesis.For the second question (is it *more* than 50%?), we also fail to reject the null hypothesis.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Set up the hypotheses
2. Calculate $\hat{p}$
3. Calculate the test statistic
4. Calculate the P-value
5. Draw a conclusion for the first question
6. Draw a conclusion for the second question

Our null hypothesis $H_0$ is that the toast lands buttered side down 50% of the time.
So, $H_0: p = 0.5$ .

For the first question, we're testing if it's different from 50%, so our alternative hypothesis $H_1$ is $H_1: p \neq 0.5$ .
This is a two-tailed test.

We have 21 out of 46 toasts landing buttered side down.
So, our sample proportion $\hat{p}$ is $\hat{p} = \frac{21}{46} \approx 0.4565$ .

The test statistic is given by $z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$ , where $p_0$ is the value of $p$ under the null hypothesis (0.5), and $n$ is the sample size (46).

Since this is a two-tailed test, we need to find the area in both tails.
First, let's find the area to the left of $z = -0.591$ .
Using a z-table or calculator, we find this area to be approximately 0.277.

Since it's a two-tailed test, we multiply this value by 2 to get the P-value: $P \approx 2 \cdot 0.277 = 0.554$ .

Our P-value (0.554) is greater than our significance level (0.10).

Therefore, we fail to reject the null hypothesis.
There's not enough evidence to say that the toast doesn't land buttered side down 50% of the time.

For the second question, the alternative hypothesis would be $H_1: p > 0.5$ , making it a right-tailed test.

The P-value for a one-tailed test would be approximately $1 - 0.277 = 0.723$ .
This is still greater than 0.10.

So, we again fail to reject the null hypothesis.
There's not enough evidence to conclude that the toast lands buttered side down more than 50% of the time.

For the first question (is it different from 50%?), we fail to reject the null hypothesis.
For the second question (is it more than 50%?), we also fail to reject the null hypothesis.