Hypothesis Test for a Population Proportion
Many investors and financial analysts believe the Dow Jones Industrial Average (DJIA) barometer of the overall stock market. On January \( 31,2006,9 \) of the 30 stocks makins increased in price (The Wall Street Journal, February 1, 2006). On the basis of this fac claims we can assume that \( 30 \% \) of the stocks traded on the New York Stock Exchange ( 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 significar You use a significance level of \( \alpha=0.01 \).
What is the test statistic for this sample? (Report answer accurate to three decimal plac test statistic \( = \)
What is the \( p \)-value for this sample? (Report answer accurate to four decimal places.) \( p \)-value \( = \)
The \( \mathrm{p} \)-value is...
less than (or equal to) \( \alpha \)
greater than \( \alpha \)
Answer
5. Compare $p$-value to the given significance level ($\alpha=0.01$): Less than (or equal to) $\alpha$ or Greater than $\alpha$
Step 1 ：1. Calculate the sample proportion ($\hat{p}$) and its complementary proportion ($1-\hat{p}$): $\hat{p}=\frac{24}{58}$, $1-\hat{p}=\frac{34}{58}$
Step 2 ：2. Calculate the standard error (SE) using the assumed population proportion ($p_0 = 0.3$) and sample size (n=58): $\mathrm{SE}=\sqrt{\frac{p_0(1-p_0)}{n}}=\sqrt{\frac{0.3(1-0.3)}{58}}$
Step 3 ：3. Calculate the test statistic: $z=\frac{\hat{p}-p_0}{\mathrm{SE}}=\frac{\frac{24}{58}-0.3}{\sqrt{\frac{0.3(1-0.3)}{58}}}$
Step 4 ：4. Using a standard normal (z) distribution table, find the p-value for (${z}$): $p$-value
Step 5 ：5. Compare $p$-value to the given significance level ($\alpha=0.01$): Less than (or equal to) $\alpha$ or Greater than $\alpha$
