Final answer:
A hypothesis test was performed using the broker's sample data, comparing the sample proportion to the surveyed proportion of 48.8%. A z-score of 2.51 was calculated, which exceeds the critical z-value range for a 0.05 significance level two-tailed test, leading to the rejection of the null hypothesis, suggesting the survey might be inaccurate.
Step-by-step explanation:
To determine whether the survey can be considered accurate, we perform a hypothesis test using the broker's sample data. The null hypothesis is that the true proportion of families owning stock is equal to 48.8%, as stated by the survey. The alternative hypothesis is that the true proportion of families owning stock is different from 48.8%.
The sample proportion (p-hat) is calculated by dividing the number of families owning stock in the sample (142) by the total number of families in the sample (250), which is 0.568 (56.8%). To perform the hypothesis test, we use the standard error of the proportion, which is given by the formula SE = sqrt(p(1-p)/n), where p is the proportion from the survey and n is the sample size. In this case, SE = sqrt(0.488(1-0.488)/250) = 0.0319.
Next, we calculate the z-score, which tells us how far the sample proportion is from the hypothesized proportion in terms of standard errors. The z-score is given by (p-hat - p) / SE. For our data, the z-score is (0.568 - 0.488) / 0.0319 = 2.51.
At the 0.05 significance level, we look at the critical z-values, which are approximately -1.96 and 1.96 for a two-tailed test. Since our calculated z-score of 2.51 is outside this range, we reject the null hypothesis, meaning the survey result of 48.8% could be inaccurate based on the broker's sample.