Question

You are conducting a study to see if the probability of catching the flu this year is significantly more than 0.33. You use a significance level of alpha= 0.10

H_0 : p = 0.33

You obtain a sample of size n=167 in which there are 62 successes.

H_1 : p > 0.33

What is the test statistic for this sample? (Report answer accurate to three decimal places.)
test statistic = __________________

What is the p-value for this sample? (Report answer accurate to four decimal places.)
p-value = __________________

Answer 1

To calculate the test statistic and p-value for the given problem, we will follow the steps outlined below.

### Step 1: Calculate the Sample Proportion
The sample proportion (\( \hat{p} \)) is calculated as follows:

\[
\hat{p} = \frac{x}{n}
\]

Where:
- \( x = 62 \) (number of successes)
- \( n = 167 \) (sample size)

So,

\[
\hat{p} = \frac{62}{167} \approx 0.3713
\]

### Step 2: Calculate the Standard Error (SE)
The standard error of the sample proportion can be calculated using the formula:

\[
SE = \sqrt{\frac{p_0(1 - p_0)}{n}}
\]

Where \( p_0 = 0.33 \) is the population proportion under the null hypothesis.

Calculating \( SE \):

\[
SE = \sqrt{\frac{0.33 \times (1 - 0.33)}{167}} = \sqrt{\frac{0.33 \times 0.67}{167}} = \sqrt{\frac{0.2211}{167}} \approx \sqrt{0.001324} \approx 0.0364
\]

### Step 3: Calculate the Test Statistic
The test statistic (\( z \)) is calculated using the formula:

\[
z = \frac{\hat{p} - p_0}{SE}
\]

Substituting the values we have:

\[
z = \frac{0.3713 - 0.33}{0.0364} \approx \frac{0.0413}{0.0364} \approx 1.136
\]

So the test statistic is approximately:

\[
\text{test statistic} = 1.136
\]

### Step 4: Calculate the p-value
Since we are conducting a one-tailed test (right-tailed), we will find the p-value corresponding to the calculated \( z \) statistic from the standard normal distribution. The p-value can be found using a z-table or a calculator.

Using standard normal distribution tables (or a calculator), we find:

1. Find the area to the left of \( z = 1.136 \).
2. Subtract this value from 1 to get the area to the right (p-value).

Using a z-table or calculator, we find:

\[
P(Z < 1.136) \approx 0.8714
\]

Therefore,

\[
\text{p-value} = 1 - 0.8714 = 0.1286
\]

### Final Answers
Thus, the final results are:

- Test statistic: **1.136**
- p-value: **0.1286**

**Test statistic = 1.136**
**p-value = 0.1286**