Question

Below are three different hypothesis tests about population proportions. For each test, use StatKey and the information given to calculate the appropriate p-value and make the correct conclusion.
(1) H0: p = 0.3 vs Ha: p ? 0.3. In their survey, they had a count of 38 using a sample size n=100.

1.a) What is p-hat for this sample?
Using StatKey, generate a randomization distribution using at least 4000 samples. Remember to select Edit Data to input sample information, and to edit the null hypothesis.
1.b) What is the p-value using this randomization distribution?
1.c) At a significance level of 0.05, what is the conclusion for this hypothesis test?

(2) H0: p = 0.7 vs Ha: p ? 0.7. In their survey, they had a count of 320 using a sample size n=500.

2.a) What is p-hat for this sample?
Using StatKey, generate a randomization distribution using at least 4000 samples. Remember to select Edit Data to input sample information, and to edit the null hypothesis.
2.b) What is the p-value using this randomization distribution?
2.c) At a significance level of 0.05, what is the conclusion for this hypothesis test?

(3) H0: p = 0.6 vs Ha: p < 0.6. In their survey, they had a count of 110 using a sample size n=200.

3.a) What is p-hat for this sample?
Using StatKey, generate a randomization distribution using at least 4000 samples. Remember to select Edit Data to input sample information, and to edit the null hypothesis.
3.b) What is the p-value using this randomization distribution?
3.c) At a significance level of 0.05, what is the conclusion for this hypothesis test?

Answer 1

Answer:

1a) p-hat=0.38

1b) P=0.08

1c) The null hypothesis is not rejected

2a) p-hat=0.64

2b) P=0.0027

2c) The null hypothesis is rejected

3a) p-hat=0.55

3b) P=0.153

3c) The null hypothesis is not rejected

Step-by-step explanation:

(1) H0: p = 0.3 vs Ha: p ≠ 0.3. In their survey, they had a count of 38 using a sample size n=100.

1a) The p-hat is p-hat=38/100=0.38.

1b) The standard deviation is

$\sigma=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.3*0.7}{100}}=0.046$

The sample size is n=100.

The z-value is:

$z=\frac{\hat{p}-p}{\sigma}=\frac{0.38-0.3}{0.046}=\frac{0.08}{0.046}= 1.74$

As it is a two-sided test, the p-value considers both tails of the distribution.

The p-value for this |z|=1.74 is P=0.08.

1c) The null hypothesis is not rejected.

(2) H0: p = 0.7 vs Ha: p ≠ 0.7. In their survey, they had a count of 320 using a sample size n=500.

2a) The p-hat is p-hat=320/500=0.64.

2b) The standard deviation is

$\sigma=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.7*0.3}{500}}=0.02$

The sample size is n=500.

The z-value is:

$z=\frac{\hat{p}-p}{\sigma}=\frac{0.64-0.7}{0.02}=\frac{-0.06}{0.02}=-3$

As it is a two-sided test, the p-value considers both tails of the distribution.

The p-value for this |z|=3 is P=0.0027.

2c) The null hypothesis is rejected.

(3) H0: p = 0.6 vs Ha: p < 0.6. In their survey, they had a count of 110 using a sample size n=200.

2a) The p-hat is p-hat=110/200=0.55.

2b) The standard deviation is

$\sigma=\sqrt{\frac{p(1-p)}{n}}=\sqrt{\frac{0.6*0.4}{200}}=0.035$

The sample size is n=200.

The z-value is:

$z=\frac{\hat{p}-p}{\sigma}=\frac{0.55-0.6}{0.035}=\frac{-0.05}{0.035}=-1.43$

As it is a two-sided test, the p-value considers both tails of the distribution.

The p-value for this |z|=1.43 is P=0.153.

2c) The null hypothesis is not rejected.