Question

QUESTION PART A: You want to obtain a sample to estimate a population proportion. At this point in time, you have no reasonable estimate for the population proportion. You would like to be 99.9% confident that you esimate is within 1.5% of the true population proportion. How large of a sample size is required?

n =

QUESTION PART B: If n = 300 and ˆp (p-hat) = 0.3, construct a 90% confidence interval.

Give your answers to three decimals

_____< p < _____

QUESTION PART C: 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 53 stocks traded on the NYSE that day showed that 26 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 α=0.02α=0.02.

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) αα
• greater than αα

This test statistic leads to a decision to...

• reject the null
• accept the null
• fail to reject the null

As such, the final conclusion is that...

• There is sufficient evidence to warrant rejection of the claim that the proportion of stocks that went up is is more than 0.3.
• There is not sufficient evidence to warrant rejection of the claim that the proportion of stocks that went up is is more than 0.3.
• The sample data support the claim that the proportion of stocks that went up is is more than 0.3.
• There is not sufficient sample evidence to support the claim that the proportion of stocks that went up is is more than 0.3.

Answer 1

1) At 99.9% confidence level, the critical value is z^* = 3.27

Margin of error = 0.015

Or, z^* * sqrt(p(1 - p)/n) = 0.015

Or, 3.27 * sqrt((0.5 * 0.5)/n) = 0.015

Or, n = (3.27 * sqrt(0.5 * 0.5)/0.015)^2

Or, n = 11881

2) At 90% confidence level, the critical value is z_0.05 = 1.645

The 90% confidence interval is

= 0.256, 0.344

0.256 < p < 0.344

3) H₀: p = 0.3

H₁: p > 0.3

= 26/53 = 0.491

The test statistic is

  

= 3.034

P-value = P(Z > 3.034)

= 1 - P(Z < 3.034)

= 1 - 0.9988

= 0.0012

The P-value is less than .

Reject the null

There is not sufficient evidence to warrant rejection of the claim that the proportion of stocks that went up is more than 0.3.