Question

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. Formulate null and alternative hypotheses to test the analyst’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 α
α= .01 as the level of significance. What is your
conclusion?

Answer 1

a) The null and alternative hypothesis are:

`H_0: \pi=0.3\\\\H_a:\pi\\eq 0.3`

b) The point estimate for the true proportion is the sample proportion, and has a value of p=0.48.

`p=X/n=24/50=0.48`

c) The conclusion is that there is enough evidence to support the claim that the proportion of stocks that went up the same day differs from 30%.

Explanation:

This is a hypothesis test for a proportion.

The claim is that the proportion of stocks that went up the same day differs from 30%.

Then, the null and alternative hypothesis are:

`H_0: \pi=0.3\\\\H_a:\pi\\eq 0.3`

The significance level is 0.01.

The sample has a size n=50.

The point estimate for the true proportion is the sample proportion, and has a value of p=0.48.

`p=X/n=24/50=0.48`

The standard error of the proportion is:

`\sigma_p=\sqrt{(\pi(1-\pi))/(n)}=\sqrt{(0.3*0.7)/(50)}\\\\\\ \sigma_p=√(0.004)=0.065`

Then, we can calculate the z-statistic as:

`z=(p-\pi+0.5/n)/(\sigma_p)=(0.48-0.3-0.5/50)/(0.065)=(0.17)/(0.065)=2.623`

This test is a two-tailed test, so the P-value for this test is calculated as:

`P-value=2\cdot P(z>2.623)=0.009`

As the P-value (0.009) is smaller than the significance level (0.01), the effect is significant.

The null hypothesis is rejected.

There is enough evidence to support the claim that the proportion of stocks that went up the same day differs from 30%.