Many investors and financial analysts believe the Dow Jones Industrial Average (DJIA) gives a good barometer of the overall stock market. On January 31, 2017, 9 of the 30 stocks making up the DJIA increased in price. On the basis of this fact, a financial analyst claims we can assume that $51 \%$ of the stocks traded on the New York Stock Exchange (NYSE) went up the same day. A sample of 66 stocks traded on the NYSE that day showed that 38 went up. You are conducting a study to see if the proportion of stocks that went up is significantly different from $51 \%$. You use a significance level of $\alpha=0.05$.
1. For this study, we should use
Hypothesis Test for a Population Proportion
$0^{\circ}$
2. The null and alternative hypotheses would be:
3. The test statistic $=$ (please show your answer to 3 decimal places.)
4. The p-value $=$ (Please show your answer to 4 decimal places.)
5. The $p$-value is $> \rightarrow \checkmark \alpha$

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 \(> \rightarrow \checkmark \alpha\)