​Claim: Fewer than 91
​%
of adults have a cell phone. In a reputable poll of 1104

​adults, 84
​%
said that they have a cell phone. Find the value of the test statistic.
Question content area bottom
Part 1
The value of the test statistic is negative 8.10
.

We are tasked with determining the test statistic for the given claim. Let's break this down step by step.

### Information provided:
1. Claim: **Fewer than 91%** of adults have a cell phone. This is a **left-tailed** test.
2. Sample size (\(n\)): 1104 adults.
3. Sample proportion (\( \hat{p} \)): 84% or 0.84.
4. Null hypothesis proportion (\( p_0 \)): 91% or 0.91.
5. Test statistic formula for proportions:
   \[
   z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}}
   \]

### Steps:

1. **Identify the null and alternative hypotheses**:
   - \( H_0 \): \( p = 0.91 \) (the proportion is 91%)
   - \( H_a \): \( p < 0.91 \) (fewer than 91%)

2. **Calculate the standard error (SE)**:
   \[
   SE = \sqrt{\frac{p_0(1 - p_0)}{n}} = \sqrt{\frac{0.91(1 - 0.91)}{1104}} = \sqrt{\frac{0.091}{1104}}
   \]

3. **Calculate the test statistic (z)**:
   \[
   z = \frac{\hat{p} - p_0}{SE} = \frac{0.84 - 0.91}{SE}
   \]

Now, let's compute the values.

I'll calculate that for you.The value of the test statistic is approximately **-8.13**, which matches closely with the provided test statistic value of **-8.10**.

Would you like more details on the calculation or have any other questions?

Here are some related questions for further exploration:
1. What does a negative test statistic indicate in hypothesis testing?
2. How do we interpret a left-tailed hypothesis test?
3. What is the p-value associated with this test statistic, and how do we find it?
4. How does sample size affect the standard error in proportion tests?
5. When should we reject the null hypothesis based on the test statistic?

**Tip**: A highly negative test statistic suggests strong evidence against the null hypothesis, indicating that the sample proportion is significantly lower than the hypothesized proportion.

Claim: Fewer than 91% of adults have a cell phone. In a reputable poll of 1104 adults, 84% said that they have a cell phone. Find the value of the test statistic.

Learn how to calculate the test statistic for a claim that fewer than 91% of adults have a cell phone, using a sample size of 1104 and a sample proportion of 84%. This problem explores hypothesis testing, the Z-test for proportions, and calculating the standard error, suitable for undergraduate statistics students.

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