A certain article indicates that in a sample of 1,000 dog owners, 680 said that they take more pictures of
their dog than of their significant others or friends, and 440 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 a 90% confidence interval for the proportion of dog owners who take more pictures of their
dog than of their significant others or friends. (Enter your answer using interval notation. Round your
numerical values to three decimal places.)
(b) Construct a 95% confidence interval for the proportion of dog owners who are more likely to complain
to their dog than to a friend. (Enter your answer using interval notation. Round your numerical values
to three decimal places.)
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(a) To construct a 90% confidence interval for the proportion of dog owners who take more pictures of their dog than of their significant others or friends, we can use the formula for the confidence interval:
CI = p̂ ± Z * √(p̂(1-p̂)/n)
where p̂ is the sample proportion, Z is the Z-score corresponding to the desired confidence level (90% in this case), and n is the sample size.
In this case, p̂ = 680/1000 = 0.680, Z = 1.645 (corresponding to a 90% confidence level), and n = 1000.
Plug these values into the formula:
CI = 0.680 ± 1.645 * √(0.680(1-0.680)/1000)
CI = 0.680 ± 1.645 * √(0.680 * 0.320 / 1000)
CI = 0.680 ± 1.645 * √(0.2176 /1000)
CI = 0.680 ± 1.645 * 0.01473
CI = 0.680 ± 0.0242
CI = (0.656, 0.704)
The 90% confidence interval for the proportion of dog owners who take more pictures of their dog than of their significant others or friends is (0.656, 0.704).
(b) To construct a 95% confidence interval for the proportion of dog owners who are more likely to complain to their dog than to a friend, we can use the same formula as above, but with a different Z-score corresponding to the 95% confidence level.
In this case, p̂ = 440/1000 = 0.440, Z = 1.96 (corresponding to a 95% confidence level), and n = 1000.
Plug these values into the formula:
CI = 0.440 ± 1.96 * √(0.440(1-0.440)/1000)
CI = 0.440 ± 1.96 * √(0.440 * 0.560 / 1000)
CI = 0.440 ± 1.96 * √(0.2464 /1000)
CI = 0.440 ± 1.96 * 0.0157
CI = 0.440 ± 0.0307
CI = (0.4093, 0.4707)
The 95% confidence interval for the proportion of dog owners who are more likely to complain to their dog than to a friend is (0.4093, 0.4707).
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