Question

The article “Most Dog Owners Take More Pictures of Their Pet Than Their Spouse” (August 22, 2016,retrieved May 6, 2017)indicates that in a sample of 1000 dog owners, 650 said that they take more pictures of their dog than their significant others or friends, and 460 said that they are more likely to complain to their dog than to a friend. Suppose that it is reasonable to consider this sample as representative of the population of dog owners.

a.Construct and interpret two different confidence interval (90% and 95%) for the proportion of dog owners who take more pictures of their dog than of their significant others or friends.

b.Which confidence interval is wider? Why?

Answer 1

total dog owners, n = 1000

Proportion of dog owner who take pictures of their dog than of their significant others or friends, = 650/1000 = 0.65

a) At = 1-0.90 = 0.10, two tailed critical value, = 1.645

90% confidence interval:

At = 1-0.95 = 0.05, two tailed critical value, = 1.96

95% confidence interval:

b) 95% confidence interval is wider. Because increase the confidence level increase the length of the confidence interval.

Answer 2

To construct confidence intervals for the proportion of dog owners who take more pictures of their dogs than their significant others or friends, we can use the sample proportions and the appropriate formulas.

Let's define: n = Total number of dog owners in the sample = 1000 x = Number of dog owners who take more pictures of their dogs than significant others or friends = 650

(a) Confidence Interval at 90% confidence level:

The formula for calculating the confidence interval for a proportion is:

CI = p̂ ± Z * sqrt((p̂ * (1 - p̂)) / n)

where: p̂ is the sample proportion (x/n), Z is the Z-score for the desired confidence level (for 90% confidence level, Z ≈ 1.645), n is the sample size.

Calculating the confidence interval:

p̂ = 650/1000 = 0.65 Z (for 90% confidence) ≈ 1.645 n = 1000

CI = 0.65 ± 1.645 * sqrt((0.65 * (1 - 0.65)) / 1000)

CI ≈ 0.65 ± 0.0248

CI ≈ (0.6252, 0.6748)

Interpretation: We are 90% confident that the true proportion of dog owners who take more pictures of their dogs than their significant others or friends is between 62.52% and 67.48%.

Confidence Interval at 95% confidence level:

The Z-score for 95% confidence level is approximately 1.96. Let's calculate the interval:

CI = 0.65 ± 1.96 * sqrt((0.65 * (1 - 0.65)) / 1000)

CI ≈ 0.65 ± 0.0257

CI ≈ (0.6243, 0.6757)

Interpretation: We are 95% confident that the true proportion of dog owners who take more pictures of their dogs than their significant others or friends is between 62.43% and 67.57%.

(b) The confidence interval at 95% confidence level is wider than the confidence interval at 90% confidence level. This is because a higher confidence level requires a larger margin of error to provide a more precise estimate of the true population proportion. As the confidence level increases, the Z-score also increases, resulting in a wider interval. The trade-off is that a higher confidence level means we can be more certain that the true population proportion falls within the interval.