Question

In a sample of 50 men, 33 said they used seat belts. In a sample of 64 women, 48 said they used seatbelts.

Step 1: Calculate the margin of error of a 90% confidence interval for the true difference between the proportions of men and women using a seat belt.

Step 2: Construct a 90% confidence interval for the true difference between the proportions of men and women using seat belts.

Answer 1

Solution :

Step 1 :

The margin of error for a 90% confidence interval for the true difference between two population proportions is given as follows :

Where,

and p̂_{​​​​​1}, p̂_{​​​​​2} are sample proportions, n_{​​​​​​1}, n_{​​​​​​2} are sample sizes.

Sample proportion of men who used seat belts is,

Sample proportion of women who used seat belts is,

n_{​​​​​​1} = 50, n_{​​​​​​2} = 64

Q = 1 - 0.7105 = 0.2895

Using Z-table we get, Z_(0.10/2) = 1.645

Hence, the margin of error is,

The margin of error of a 90% confidence interval for the true difference between the proportions of men and women using a seat belt is 0.1408.

Step 2 :

The 90% confidence interval for the true difference between the proportions of men and women using a seat belt is given as follows :

The 90% confidence interval for the true difference between the proportions of men and women using a seat belt is (-0.2308, 0.0508).