Two Sample Population Confidence Interval:
Before an intensive TV advertising campaign, the producers of Nike athletic
shoes find that 30 of a random sample of 217 upper-income adults were aware
of their new leisure shoe line. A second random sample of 352 such adults
is taken after the campaign. Now 122 of the persons sampled can identity the
new line. Give a 99% confidence interval for the increase in the proportion of
upper income adults showing brand awareness.
All of the confidence interval qustions can be answered by reading the Wikepieda article cited in the last question above.
To calculate a confidence interval for the increase in the proportion of upper-income adults showing brand awareness, we need to compare the proportions of awareness before and after the TV advertising campaign.
First, let's calculate the proportion of awareness before the campaign. In the initial random sample of 217 upper-income adults, 30 were aware of the new leisure shoe line. Therefore, the proportion of awareness before the campaign is:
p1 = 30/217 ≈ 0.1384
Next, let's calculate the proportion of awareness after the campaign. In the second random sample of 352 upper-income adults, 122 were aware of the new line. So, the proportion of awareness after the campaign is:
p2 = 122/352 ≈ 0.3466
To calculate the confidence interval, we use the formula:
p̂1 - p̂2 ± Z * √[(p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2)]
Where:
- p̂1 and p̂2 are the sample proportions
- Z is the Z-score corresponding to the desired confidence level (99% in this case)
- n1 and n2 are the sample sizes
Now, let's substitute the values into the formula:
p̂1 - p̂2 ± Z * √[(p̂1 * (1 - p̂1) / n1) + (p̂2 * (1 - p̂2) / n2)]
= 0.1384 - 0.3466 ± Z * √[(0.1384 * (1 - 0.1384) / 217) + (0.3466 * (1 - 0.3466) / 352)]
To find the Z-score corresponding to a 99% confidence level, we can look it up in the Z-table or use statistical software. The Z-score for a 99% confidence level is approximately 2.576.
Substituting the values:
0.1384 - 0.3466 ± 2.576 * √[(0.1384 * (1 - 0.1384) / 217) + (0.3466 * (1 - 0.3466) / 352)]
Now, we can calculate the confidence interval:
0.1384 - 0.3466 ± (2.576 * 0.0263)
= -0.2082 ± 0.0676
Therefore, the 99% confidence interval for the increase in the proportion of upper-income adults showing brand awareness is approximately (-0.276, -0.140).