In a random sample 500 people eating lunch at a cafeteria on various Fridays, it was found that 160 preferred seafood. Find 99% confidence interval for the actual proportion of the people who eat seafood on Fridays at this cafeteria.Give the upper limit.
To find the 99% confidence interval for the actual proportion of people who eat seafood on Fridays at this cafeteria, we can use the formula:
CI = p̂ ± z * √(p̂(1-p̂)/n)
Where:
CI = Confidence Interval
p̂ = Sample Proportion
z = Z-score corresponding to the desired confidence level
n = Sample Size
First, calculate the sample proportion:
p̂ = 160/500 = 0.32
Next, find the Z-score corresponding to the 99% confidence level. The Z-score for a 99% confidence level is approximately 2.576.
Plug the values into the formula:
CI = 0.32 ± 2.576 * √((0.32*(1-0.32))/500)
Calculating the standard deviation:
√((0.32*(1-0.32))/500) ≈ 0.023
CI = 0.32 ± 2.576 * 0.023
Calculating the confidence interval:
CI = 0.32 ± 0.0593
To find the upper limit of the confidence interval, add the margin of error to the sample proportion:
Upper Limit = 0.32 + 0.0593 = 0.3793
Therefore, the upper limit of the 99% confidence interval for the actual proportion of people who eat seafood on Fridays at this cafeteria is approximately 0.3793.
To find the upper limit of the 99% confidence interval for the proportion of people who eat seafood on Fridays at the cafeteria, we can use the formula:
Upper limit = sample proportion + margin of error
The sample proportion is the observed proportion of people who preferred seafood in the sample, which is 160 out of 500:
Sample proportion (p̂) = Number of people who preferred seafood / Sample size
= 160 / 500
= 0.32
Next, we need to calculate the margin of error. The formula for the margin of error is:
Margin of error = Critical value * Standard error
First, we need to find the critical value. Since we want a 99% confidence interval, we can use the Z-score for a 99% confidence level, which is 2.576.
Next, we need to calculate the standard error. The formula for the standard error of a proportion is:
Standard error (SE) = sqrt[(p̂ * (1 - p̂)) / n]
Where:
p̂ = Sample proportion
n = Sample size
Substituting the values into the formula, we get:
Standard error (SE) = sqrt[(0.32 * (1 - 0.32)) / 500]
= sqrt(0.32 * 0.68 / 500)
≈ 0.0223
Now, we can substitute the values into the margin of error formula:
Margin of error = 2.576 * 0.0223
≈ 0.0574
Finally, we can calculate the upper limit of the confidence interval:
Upper limit = Sample proportion + Margin of error
= 0.32 + 0.0574
≈ 0.3774
Therefore, the upper limit of the 99% confidence interval for the actual proportion of people who eat seafood on Fridays at this cafeteria is approximately 0.3774.