Question

5. Marketers believe that 92% of adults own a cell phone. A cell phone manufacturer believes that number is actually lower. In a sample of 200 adults, 87% own a cell phone. At the 1% significance level, determine if the proportion of adults that own a cell phone is lower than the marketers’ claim. For all the questions above: a. Formulate the hypotheses b. Determine an appropriate test statistic. c. Calculate the test statistic d. State your conclusion

Answer 1

a. Hypotheses:

-
`\(H_0: p \geq 0.92\)`(Marketers' claim)

-
`\(H_1: p < 0.92\)` (Manufacturer's belief)

b. Test Statistic:

- Use z-test for proportions.

c. Calculation:

- Calculate z using sample proportion.

d. Conclusion:

- If z < critical z-value at 1%, reject
`\(H_0\),` concluding the proportion of adults owning a cell phone is lower than marketers' claim.

a. Formulate the hypotheses:

- Null hypothesis (H0): The proportion of adults owning a cell phone is equal to or higher than the marketers' claim (p ≥ 0.92).

- Alternative hypothesis (H1): The proportion of adults owning a cell phone is lower than the marketers' claim (p < 0.92).

b. Determine an appropriate test statistic:

- Since you are dealing with proportions and comparing a sample proportion to a known population proportion, you can use the z-test for proportions.

`\[ z = \frac{\hat{p} - p_0}{\sqrt{(p_0 \cdot (1 - p_0))/(n)}} \]`

where
`\(\hat{p}\)` is the sample proportion,
`\(p_0\)`is the claimed proportion, and \(n\) is the sample size.

c. Calculate the test statistic:

`\[ z = \frac{0.87 - 0.92}{\sqrt{(0.92 \cdot (1 - 0.92))/(200)}} \]`

d. State your conclusion:

- Compare the calculated z-value to the critical z-value at a 1% significance level (usually obtained from a z-table or statistical software).

- If the calculated z-value is less than the critical z-value, reject the null hypothesis.

- In the conclusion, state whether there is enough evidence to reject the claim that the proportion of adults owning a cell phone is equal to or higher than the marketers' claim at the 1% significance level.

Note: The specific numerical values for the z-test and critical values would need to be calculated or looked up using a statistical table, and the conclusion will depend on these values.