Question

In: Statistics and Probability

In a recent survey of home-buyer preferences, consumers were asked about desired characteristics, such as a...

In a recent survey of home-buyer preferences, consumers were asked about desired characteristics, such as a wood-burning fireplace, a den/library, and flooring. In particular, each person was asked whether a separate dining room is essential. The sample size and the number who responded Yes to this question are given in the table by geographic region.

a. [11 marks, 2 each, & 1 for value of z] Find a 99% confidence interval for the true proportion of home-buyers who believe a separate dining room is essential in each geographic region. Show all your work in order to get full marks.

b. [4 marks] Which confidence interval in part (a) is the largest? Which confidence interval in part (a) is the smallest? Why? Show all your work in order to get full marks.

Geographic region	Sample size	Number who responded Yes
Northeast	225	180
Mideast	276	224
South Central	301	232
South Atlantic	454	377
West	366	304

Expert Solution

a)

Answer:

Explanation:

Northeast

The confidence interval for the proportion is obtained using the formula,

Where,

Similarly, the confidence interval is obtained for other regions. The calculations are done in excel. The screenshot is shown below,

b)

Answer:

Largest confidence interval: Northeast

Smallest confidence interval: South Atlantic

Reason: The width of the confidence interval depends of the sample size and the proportion estimate.

Explanation:

The confidence interval width is obtained by taking the difference between the upper and lower limits of the confidence interval.

Geographic region	Lower limit	Upper limit	Width
Northeast	0.7313	0.8687	0.1374
Mideast	0.7510	0.8722	0.1213
South Central	0.7084	0.8332	0.1248
South Atlantic	0.7850	0.8758	0.0907
West	0.7801	0.8811	0.1010

From the above table, we can see that the confidence interval for the region Northeast is largest and the confidence interval for the region South Atlantic is smallest.

The width of the confidence interval is defined as the margin of error as shown below,

From the above formula, we can observe that the margin of error depends on,

i) proportion p(1-p) i.e. as the proportion decreases, the margin of error increases.

ii) sample size, i.e. as the sample size decreases, the margin of error increases.

For the Northeast region, the proportion estimate and the sample size both are lower which leads to largest confidence interval.

For the South Atlantic region, the proportion estimate and the sample size both are higher which leads to the smallest confidence interval.

orchestra answered 1 year ago

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