Question

a nutritionist believes that 10% of teenagers eat cereal for breakfast. to investigate this claim, she selects a random sample of 150 teenagers and finds that 25 eat cereal for breakfast. she would like to know if the data provide convincing evidence that the true proportion of teenagers who eat cereal for breakfast differs from 10%. what are the values of the test statistic and p-value for this test?

Answer 1

To test if the data provide convincing evidence that the true proportion of teenagers who eat cereal for breakfast differs from 10%, we can perform a hypothesis test. The test statistic is approximately -2.89 and the p-value is approximately 0.003. Since the p-value is less than the significance level, we reject the null hypothesis.

Step-by-step explanation:

To test if the data provide convincing evidence that the true proportion of teenagers who eat cereal for breakfast differs from 10%, we can perform a hypothesis test.

1. Step 1: State the hypotheses:
2. Step 2: Set the significance level (alpha) to 0.05.
3. Step 3: Calculate the test statistic using the formula:
   test statistic = (x - np) / sqrt(np(1 - p))
   where x is the number of teenagers who eat cereal for breakfast (25), n is the sample size (150), and p is the assumed proportion (0.1).
4. Step 4: Determine the p-value using the test statistic and the appropriate distribution (in this case, the standard normal distribution).
5. Step 5: Make a decision based on the p-value:
6. If the p-value is less than the significance level (0.05), reject the null hypothesis.
7. If the p-value is greater than or equal to the significance level, fail to reject the null hypothesis.

In this case, the test statistic is approximately -2.89 and the p-value is approximately 0.003. Since the p-value is less than the significance level, we reject the null hypothesis. This provides convincing evidence that the true proportion of teenagers who eat cereal for breakfast differs from 10%.

Answer 2

Based on the test statistic of 1.67 and the p-value of 0.0004, there is not enough evidence to conclude that the true proportion of teenagers who eat cereal for breakfast differs from 10%.

Step-by-step explanation:

To conduct a hypothesis test to determine if the true proportion of teenagers who eat cereal for breakfast differs from 10%, we can use a hypothesis test for proportions.

The null hypothesis (H0) is that the true proportion is 10%, and the alternative hypothesis (Ha) is that the true proportion is different from 10%.

The test statistic is calculated using the formula: z = (p_observed - p_expected) / sqrt((p_expected * (1 - p_expected)) / n), where p_observed is the proportion observed in the sample, p_expected is the hypothesized proportion (10%), and n is the sample size.

The p-value is the probability of observing a test statistic as extreme as the one obtained or more extreme, assuming that the null hypothesis is true.

In this case, the test statistic is z = (25/150 - 0.10) / sqrt((0.10 * 0.90) / 150) = 1.67, and the p-value is approximately 0.0004.