A survey of Internet users reported that 21% downloaded music onto their computers. The filing of lawsuits by the recording industry may be a reason why this percent has decreased from the estimate of 31% from a survey taken two years before. Suppose we are not exactly sure about the sizes of the samples. Perform the calculations for the significance tests and 95% confidence intervals under each of the following assumptions. (Use previous − recent. Round your test statistics to two decimal places and your confidence intervals to four decimal places.)
(i) Both sample sizes are 1000.
z = ____________
(ii) Both sample sizes are 1600.
z = _____________
(iii) The sample size for the survey reporting 31% is 1000 and the sample size for the survey reporting 21% is 1600.
z = _____________
To perform the calculations for the significance tests and 95% confidence intervals, we need to use the formula for the z-test:
z = (p̂₁ - p̂₂) / √(p̂(1 - p̂) * (1/n₁ + 1/n₂))
where:
- p̂₁ is the proportion in the first survey
- p̂₂ is the proportion in the second survey
- p̂ is the pooled proportion ((p̂₁n₁ + p̂₂n₂) / (n₁ + n₂))
- n₁ is the sample size for the first survey
- n₂ is the sample size for the second survey
Now let's calculate the z-values for each scenario:
(i) Both sample sizes are 1000.
Using the given information:
p̂₁ = 31% = 0.31
p̂₂ = 21% = 0.21
n₁ = n₂ = 1000
Now let's calculate the pooled proportion:
p̂ = (0.31 * 1000 + 0.21 * 1000) / (1000 + 1000) = 0.26
Now let's plug in the values into the formula:
z = (0.31 - 0.21) / √(0.26 * (1 - 0.26) * (1/1000 + 1/1000))
z ≈ 1.62 (round to two decimal places)
(ii) Both sample sizes are 1600.
Using the given information:
p̂₁ = 31% = 0.31
p̂₂ = 21% = 0.21
n₁ = n₂ = 1600
Now let's calculate the pooled proportion:
p̂ = (0.31 * 1600 + 0.21 * 1600) / (1600 + 1600) = 0.26
Now let's plug in the values into the formula:
z = (0.31 - 0.21) / √(0.26 * (1 - 0.26) * (1/1600 + 1/1600))
z ≈ 2.73 (round to two decimal places)
(iii) The sample size for the survey reporting 31% is 1000 and the sample size for the survey reporting 21% is 1600.
Using the given information:
p̂₁ = 31% = 0.31
p̂₂ = 21% = 0.21
n₁ = 1000
n₂ = 1600
Now let's calculate the pooled proportion:
p̂ = (0.31 * 1000 + 0.21 * 1600) / (1000 + 1600) = 0.246
Now let's plug in the values into the formula:
z = (0.31 - 0.21) / √(0.246 * (1 - 0.246) * (1/1000 + 1/1600))
z ≈ 2.67 (round to two decimal places)
To calculate the 95% confidence intervals, we can use the formula:
CI = (p̂₁ - p̂₂) ± z * √(p̂(1 - p̂) * (1/n₁ + 1/n₂))
Based on the z-values calculated above, we can calculate the confidence intervals for each scenario.