Trends in fashion have signaled the emergence of green as the color of the century. It ranked third as the most preferred color among designers having a market share of 9.24%. Its popularity may be partially due to an increased interest in the environment. Besides, “Green symbolizes harmony and counteracts emotional stress.”
a. To verify the accuracy of these figures, a survey is being considered. Find the size of the sample that you should take if you want the estimate of the true proportion of designers who preferred green not to differ by more than 0.03 with 95% confidence.
b. Due to financial constraints, only a random sample of 250 local designers were interviewed and asked about their preferred color. Results showed that 27 preferred the color green. Construct a 95% confidence interval for the true proportion of local designers who preferred the color green. Interpret your results.
c. From the results in (b), is there sufficient evidence to indicate that the proportion of local designers who preferred the color green is different from 9.24%? Justify your answer.
a. To estimate the size of the sample needed for the estimate of the true proportion of designers who preferred green not to differ by more than 0.03 with 95% confidence, we can use the formula for sample size calculation for estimating a proportion:
n = (Z^2 * p * (1-p)) / E^2
Where:
n = sample size needed
Z = Z-score corresponding to the desired confidence level (in this case, 95% confidence corresponds to a Z-score of approximately 1.96)
p = estimated proportion of designers who preferred green (9.24% or 0.0924)
E = maximum acceptable margin of error (0.03)
Plugging in the values, we can calculate the sample size:
n = (1.96^2 * 0.0924 * (1-0.0924)) / 0.03^2
n = 0.166967424 / 0.0009
n ≈ 185.52
Rounding up to the nearest whole number, the estimated sample size needed is 186.
b. To construct a 95% confidence interval for the true proportion of local designers who preferred the color green, we can use the formula for confidence intervals of proportions:
Confidence interval = sample proportion ± (Z * √(sample proportion * (1 - sample proportion) / sample size))
Where:
sample proportion = number of designers who preferred green / sample size
Z = Z-score corresponding to the desired confidence level (in this case, 95% confidence corresponds to a Z-score of approximately 1.96)
sample size = 250
number of designers who preferred green = 27
Plugging in the values, we can calculate the confidence interval:
sample proportion = 27 / 250 = 0.108
Confidence interval = 0.108 ± (1.96 * √(0.108 * (1 - 0.108) / 250))
Confidence interval = 0.108 ± (1.96 * √(0.108 * 0.892 / 250))
Confidence interval = 0.108 ± (1.96 * √(0.096816 / 250))
Confidence interval = 0.108 ± (1.96 * √0.000387264)
Confidence interval ≈ 0.108 ± (1.96 * 0.01968)
Confidence interval ≈ 0.108 ± 0.03845
The 95% confidence interval for the true proportion of local designers who preferred the color green is approximately 0.07 to 0.15. This means that, based on the sample data, we can be 95% confident that the true proportion of local designers who preferred the color green falls within this range.
c. To determine if there is sufficient evidence to indicate that the proportion of local designers who preferred the color green is different from 9.24%, we can compare the confidence interval obtained in part (b) with the estimated proportion of designers who preferred green (9.24% or 0.0924).
If the confidence interval includes the estimated proportion, then there is not sufficient evidence to indicate a difference. However, if the confidence interval does not include the estimated proportion, then there is sufficient evidence to indicate a difference.
In this case, the confidence interval is approximately 0.07 to 0.15, and it does not include 0.0924. Therefore, there is sufficient evidence to indicate that the proportion of local designers who preferred the color green is different from 9.24%.