Problem
Hypothesis Test for a Population Proportion Many investors and financial analysts believe the Dow Jones Industrial Average (DJIA) gives a good barometer of the overall stock market. On January \( 31,2006,9 \) of the 30 stocks making up the DJIA increased in price (The Wall Street Journal, February 1, 2006). On the basis of this fact, a financial analyst claims we can assume that \( 30 \% \) of the stocks traded on the New York Stock Exchange (NYSE) went up the same day. A sample of 58 stocks traded on the NYSE that day showed that 24 went up. You are conducting a study to see if the proportion of stocks that went up is is significantly more than 0.3 . You use a significance level of \( \alpha=0.01 \). What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic \( = \) What is the \( p \)-value for this sample? (Report answer accurate to four decimal places.) \( p \)-value \( = \) The p-value is... less than (or equal to) \( \alpha \) greater than \( \alpha \) \( \infty \) Type here to search
Solution
Step 1 ：n = 58, X = 24, \( \hat{p} = \frac{X}{n} = \frac{24}{58} \), p_0 = 0.3, \( \alpha = 0.01 \)
Step 2 ：test statistic \( z = \frac{(\hat{p} - p_0)}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{(\frac{24}{58} - 0.3)}{\sqrt{\frac{0.3(1-0.3)}{58}}} \)
Step 3 ：z = 0.746, p-value = 1 - P(Z \le 0.746) \approx 0.2278
