Question

2 The Civil War. A national survey conducted among a simple random sample of 1,507 adults shows that 56% of Americans think the Civil War is still relevant to American politics and political life.

(a) Conduct a hypothesis test to determine if these data provide strong evidence that the majority of the Americans think the Civil War is still relevant.

(b) Interpret the p-value in this context.

(c) Calculate a 90% confidence interval for the proportion of Americans who think the Civil War is still relevant. Interpret the interval in this context, and comment on whether or not the confidence interval agrees with the conclusion of the hypothesis test

Answer 1

Solution:

Given:  A national survey conducted among a simple random sample of 1,507 adults shows that 56% of Americans think the Civil War is still relevant to American politics and political life.

thus

Sample size = n = 1507

Sample proportion =

Part a) Conduct a hypothesis test to determine if these data provide strong evidence that the majority of the Americans think the Civil War is still relevant.

thus hypothesis H0 and H1 are:

H0: p = 0.50

Vs

H1: p > 0.50

Test statistic is:

Find P-value:

P-value = P( Z > z test statistic )

P-value = P( Z > 4.66 )

P-value = 1 - P( Z < 4.66 )

Use excel command to find P-value:

=1-NORM.S.DIST( z ,cumulative)

=1-NORM.S.DIST(4.66 , TRUE)

= 0.0000016

Thus P-value =  0.0000016

P-value = 0.0000

Since P-value = 0.0000 < 0.10 level of significance , we reject null hypothesis and conclude that: these data provide strong evidence that the majority of the Americans think the Civil War is still relevant.

Part b) Interpret the p-value in this context.

The probability of obtaining sample proportion of Americans think the Civil War is still relevant to American politics and political life is 56% or more is 0.0000.

Part c) Calculate a 90% confidence interval for the proportion of Americans who think the Civil War is still relevant.

Formula:

where

Z_c is z critical value for c = 90% confidence level.

Find Area = ( 1 + c ) / 2 = ( 1 + 0.90) / 2 = 1.90 / 2 = 0.9500

Look in z table for Area = 0.9500 or its closest area and find corresponding z value.

Area 0.9500 is in between 0.9495 and 0.9505 and both the area are at same distance from 0.9500

Thus we look for both area and find both z values

Thus Area 0.9495 corresponds to 1.64 and 0.9505 corresponds to 1.65

Thus average of both z values is : ( 1.64+1.65) / 2 = 1.645

Thus Z_c = 1.645

thus

Thus

Interpret the interval in this context, and comment on whether or not the confidence interval agrees with the conclusion of the hypothesis test

We are 90% confident that the true proportion of Americans who think the Civil War is still relevant is between the limits

Since this interval exceeds the p =0.50, the confidence interval agrees with the conclusion of the hypothesis test.