Many investors and financial analysts believe the Dow Jones...

Many investors and financial analysts believe the Dow Jones Industrial Average (DJLA) provides 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 Sireet 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. Formulate null and alternative hypotheses to test the anslyst's claim. b. A sample of 50 stocks traded on the NYSE that day showed that 24 went up. What is your point estimate of the population proportion of stocks that went up? c. Conduct your hypothesis test using $\alpha=.01$ as the level of significance. What is your conclusion?

Many investors and financial analysts believe the Dow Jones Industrial Average (DJLA) provides 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 Sireet 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. Formulate null and alternative hypotheses to test the anslyst's claim.
b. A sample of 50 stocks traded on the NYSE that day showed that 24 went up. What is your point estimate of the population proportion of stocks that went up?
c. Conduct your hypothesis test using $\alpha=.01$ as the level of significance. What is your conclusion?

Key Concepts

Null and Alternative Hypotheses

In statistical hypothesis testing, formulating the null hypothesis (H0) and the alternative hypothesis (Ha) is crucial. The null hypothesis typically represents a default or status quo claim about a population parameter – for example, that a certain proportion is a specific value – while the alternative hypothesis represents the claim being tested, indicating a deviation from the null. Clearly defining these hypotheses allows an analyst to structure the test and determine whether observed data provide sufficient evidence to reject the null hypothesis.

Point Estimate

A point estimate is a single value used as an approximation for a population parameter. In the context of estimating a population proportion, the sample proportion serves as a point estimate. It is calculated by dividing the number of successes by the total sample size, providing an immediate estimate that is used in further analysis, such as hypothesis testing or constructing confidence intervals.

Significance Level

The significance level, denoted by alpha (?), is a threshold set by the analyst that determines the maximum probability of committing a Type I error – rejecting the null hypothesis when it is actually true. It is used to define the rejection region for the test statistic. When the p-value obtained from the test is less than or equal to the significance level, the null hypothesis is rejected in favor of the alternative hypothesis.

Hypothesis Testing for a Population Proportion

Hypothesis testing for a population proportion involves using sample data to test claims about the proportion of a certain attribute in a population. It typically includes calculating a test statistic, often using a z-test for proportions when the sample size is sufficiently large, and comparing that statistic to critical values determined by the significance level. This process, combined with evaluating the p-value, allows analysts to draw conclusions about the population based on sample observations.

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