Question

When analyzing the last digits of telephone numbers in Sandtown, it was found that among 1500 randomly selected phone numbers, 225 had zero as the last digit. If the digits are selected randomly, the proportion of zeroes should be 0.1 because from probability, 0 is one of the 10 possible digits.

1. Use the p-value method with a 0.01 level of significance to test the claim that the proportion of zeroes is not equal to 0.1.

2. Use the sample data to construct a 99% confidence interval to estimate the proportion of zeroes. What does the confidence interval suggest about the claim that the proportion of zeroes is 0.1?

3. Compare the results from the hypothesis test and confidence interval. Do they lead to the same conclusion? Why or why not?

Answer 1

Solution:

Given:

x = 225 , n = 1500

Population proportion = p = 0.10

Part 1) Use the p-value method with a 0.01 level of significance to test the claim that the proportion of zeroes is not equal to 0.1.

Step 1) State H0 and H1:

Step 2) Find test statistic value:

where

Thus

Step 3) Find p-value:

p-value= 2 x P( Z > 6.45)

p-value= 2 x [ 1 - P(   Z < 6.45) ]

From z table , as P( Z < 3.49) = 0.9998, then P( Z < 6.45) is approximately = 1.0000

p-value= 2 x [ 1 - 1.0000 ]

p-value = 2 x 0.0000

p-value = 0.0000

Since p-value = 0.0000 < 0.01 level of significance , we reject H0 and conclude that there is sufficient evidence to support the claim that the proportion of zeroes is not equal to 0.1.

Part 2) Use the sample data to construct a 99% confidence interval to estimate the proportion of zeroes.

Formula:

where

Z_c is z critical value for c = 0.99 confidence level.

Find Area = ( 1+c)/2 = ( 1 + 0.99 ) / 2 = 1.99 /2 = 0.9950

Thus look in z table for Area = 0.9950 or its closest area and find corresponding z critical value.

From above table we can see area 0.9950 is in between 0.9949 and 0.9951 and both are at same distance from 0.9950, Hence corresponding z values are 2.57 and 2.58

Thus average of both z values is 2.575

Thus Z_c = 2.575

Thus

Thus a 99% confidence interval to estimate the proportion of zeroes is:

What does the confidence interval suggest about the claim that the proportion of zeroes is 0.1?

a 99% confidence interval suggest that the proportion of zeroes is not equal to 0.1.

Part 3) Compare the results from the hypothesis test and confidence interval. Do they lead to the same conclusion? Why or why not?

Since p value is less than 0.01 level of significance and confidence interval is greater than p = 0.10, the results from the hypothesis test and confidence interval are same and they lead to the same conclusion.