Question

QUESTION PART A: 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 61 stocks traded on the NYSE that day showed that 6 went up.

You are conducting a study to see if the proportion of stocks that went up is is significantly less than 0.3. You use a significance level of α=0.01α=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) 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 less than 0.3.
• There is not sufficient evidence to warrant rejection of the claim that the proportion of stocks that went up is is less than 0.3.
• The sample data support the claim that the proportion of stocks that went up is is less than 0.3.
• There is not sufficient sample evidence to support the claim that the proportion of stocks that went up is is less than 0.3.

QUESTION PART B: 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 2.5% of the true population proportion. How large of a sample size is required?

n =



Do not round mid-calculation. However, use a critical value accurate to three decimal places.

(I have posted this three different times and it has been wrong, so please!! it will help me a lot)

Answer 1

Part A : A sample of 61 stocks traded on the NYSE that day showed that 6 went up. You are conducting a study to see if the proportion of stocks that went up is is significantly less than 0.3.

using a significance level of a = 0.01

Solution :

* Test statistic:

X = 6

n = 61

p̂ = x/n

​ = 6/61

= 0.098

p = 0.30

q = 1 - p = 0.70

Test statistic,Z :

Z = (p̂ - p)/√(pq/n)

Z = (0.098 – 0.30)/√(0.30*0.70/61)

= -0.202/0.058

Z = -3.4827

Test statistic = -3.483

* P-value = 0.0003 (from z table)

P-value < α , i.e (0.0003< 0.01)

* Thus, this test statistic leads to the decision that we reject the null hypothesis.

* 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 less than 0.3.

Part B:

Sample size for 99.9% confidence interval, without p̂

E = 2.5% =0.025

The critical value at Z_{​​​​​a/2} = Z0.0005 = 3.29 (from z table)

n = p̂(1 - p̂)(Z0.0005/E)^2

= 0.5(1-0.5)(3.291/0.025)^2

= 0.25(131.64)^2

= 4332.27

n = 4333

As n has integer, we take the ceiling of the above number and we get n =4333.