Question

It has been found that 51.6% of U.S. households own stocks and mutual funds. A random sample of 300 heads of households indicated that 175 owned some type of stock. At what level of significance would you conclude that this was a significant difference? Use the P-value method.

Answer 1

To determine if there is a significant difference between the sample and population proportions of households owning stocks, we use the P-value method for hypothesis testing at a 5 percent level of significance. If the P-value is less than 0.05, we reject the null hypothesis, indicating a significant difference.

Step-by-step explanation:

To determine if the observed proportion of households that owned stock in the sample is significantly different from the given population proportion, we can perform a hypothesis test using the P-value method. The null hypothesis (H0) is that the sample proportion is the same as the population proportion, while the alternative hypothesis (Ha) suggests a significant difference.

We calculate the test statistic using the formula for the proportion Z-test and then find the P-value, which is the probability of observing a test statistic as extreme as the one calculated, under the assumption that the null hypothesis is true.

If the P-value is less than the chosen level of significance (alpha), we reject the null hypothesis. In this case, alpha = 0.05 (5 percent) is often used as a standard level of significance.

To conduct the test:

1. State the null hypothesis (H0: p = 0.516) and the alternative hypothesis (Ha: p ≠ 0.516).
2. Calculate the test statistic using the sample data (175 out of 300 households own stock).
3. Determine the P-value associated with the calculated test statistic.
4. Compare the P-value to alpha = 0.05.
5. Draw a conclusion: if P-value alpha, reject H0; otherwise, do not reject H0.

If the P-value is indeed less than alpha, it suggests that there is evidence to support that the proportion of households owning stocks is different from 51.6%.