Step 1 :We are conducting a hypothesis test for a population proportion. The null hypothesis is that the proportion of stocks that went up is 51%, and the alternative hypothesis is that the proportion is not 51%.
Step 2 :We are given a sample of 66 stocks, of which 38 went up. We can use this information to calculate the sample proportion and the test statistic.
Step 3 :The test statistic is a z-score, which we can calculate using the formula for the test statistic for a population proportion.
Step 4 :Once we have the test statistic, we can calculate the p-value, which is the probability of observing a test statistic as extreme as the one we calculated, assuming the null hypothesis is true.
Step 5 :If the p-value is less than the significance level of 0.05, we reject the null hypothesis. If the p-value is greater than or equal to 0.05, we do not reject the null hypothesis.
Step 6 :Using the given data, we find that the test statistic is approximately \(1.069\) and the p-value is approximately \(0.2852\).
Step 7 :Since the p-value is greater than the significance level of \(0.05\), we do not reject the null hypothesis.
Step 8 :Therefore, we do not have sufficient evidence to conclude that the proportion of stocks that went up is significantly different from \(51 \%\).
Step 9 :So, the final answers are: The test statistic \(= \boxed{1.069}\), The p-value \(= \boxed{0.2852}\), The p-value is \(> \rightarrow \checkmark \alpha\)