1) A sample of 49 observations is taken from a normal population. The sample mean is 55 and the sample standard deviation is 10. Determine the 99% confidence interval for the population mean.
2) The Fox TV network is considering replacing one of its primetime crime investigation shows with a new family-oriented comedy show. Before a final decision is made, network executives commission a sample of 400 viewers. After viewing the comedy, 250 indicated they would watch the new show and suggested it replace the crime investigation show.
A) Estimate the value of the population proportion.
B) Compute the standard error of the proportion.
C) Develop a 99% CI for the population proportion.
Interpret your findings.
1) Formula using 99% confidence interval:
CI99 = mean ± (2.58)(sd/√n)
Note: ± 2.58 represents 99% confidence interval.
mean = 55
sd = 10
n = 49
Plug the values into the formula and calculate.
2) Formula for 99% interval estimate of the population proportion:
CI95 = p + or - 2.58[√(pq/n]
p = 250/400
q = 1 - p
n = 400
Convert the fractions to decimals, substitute the decimals into the formula, then finish the calculation.
A) and B) can be answered using the information above.
I hope this will help get you started.
For 2), the formula should read:
CI99 = p + or - 2.58[√(pq/n)]
Sorry for any confusion.
1) To determine the 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean +/- (Critical Value * (Sample Standard Deviation / Square Root of Sample Size))
First, we need to find the critical value for a 99% confidence level. The critical value can be obtained from the t-distribution table or using statistical software. For a 99% confidence level with 48 degrees of freedom (n - 1), the critical value is approximately 2.680.
Plugging in the values, we have:
Confidence Interval = 55 +/- (2.680 * (10 / √49))
Confidence Interval = 55 +/- (2.680 * 1.428)
Calculating the upper and lower limits of the confidence interval:
Upper Limit = 55 + (2.680 * 1.428)
Lower Limit = 55 - (2.680 * 1.428)
Therefore, the 99% confidence interval for the population mean is (52.96, 57.04).
2)
A) To estimate the value of the population proportion, we divide the number of viewers who indicated they would watch the new show (250) by the total sample size (400):
Population Proportion = 250 / 400
Population Proportion = 0.625
B) To compute the standard error of the proportion, we use the formula:
Standard Error = √((Population Proportion * (1 - Population Proportion)) / Sample Size)
Plugging in the values:
Standard Error = √((0.625 * (1 - 0.625)) / 400)
Calculating the standard error:
Standard Error = √((0.625 * 0.375) / 400)
Standard Error = √(0.234375 / 400)
Standard Error ≈ 0.0242
C) To develop a 99% confidence interval for the population proportion, we can use the formula:
Confidence Interval = Sample Proportion +/- (Critical Value * Standard Error)
First, we need to find the critical value for a 99% confidence level. The critical value corresponds to the z-score and can be obtained from the standard normal distribution table or using statistical software. For a 99% confidence level, the critical value is approximately 2.576.
Plugging in the values, we have:
Confidence Interval = 0.625 +/- (2.576 * 0.0242)
Calculating the upper and lower limits of the confidence interval:
Upper Limit = 0.625 + (2.576 * 0.0242)
Lower Limit = 0.625 - (2.576 * 0.0242)
Therefore, the 99% confidence interval for the population proportion is approximately (0.574, 0.676).
Interpretation: We are 99% confident that the true proportion of viewers who would watch the new show and suggest replacing the crime investigation show lies between 0.574 and 0.676.
1) To determine the 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean +/- (Z * (Sample Standard Deviation / sqrt(n)))
Where:
- Sample Mean is the mean of the sample (55).
- Z is the Z-score associated with the desired confidence level. For a 99% confidence level, the Z-score is approximately 2.576.
- Sample Standard Deviation is the standard deviation of the sample (10).
- n is the sample size (49).
Plugging in the values into the formula, we have:
Confidence Interval = 55 +/- (2.576 * (10 / sqrt(49)))
Calculating the square root of 49, we get:
Confidence Interval = 55 +/- (2.576 * (10 / 7))
Simplifying further:
Confidence Interval = 55 +/- (2.576 * 1.429)
Confidence Interval = 55 +/- 3.687
Therefore, the 99% confidence interval for the population mean is approximately (51.313, 58.687).
2) A) To estimate the value of the population proportion, we can use the formula:
Population Proportion = Sample Proportion
In this case, the sample proportion is the ratio of viewers who indicated they would watch the new show and suggested it replace the crime investigation show. Therefore:
Population Proportion = 250/400 = 0.625
So, the estimated value of the population proportion is 0.625.
B) To compute the standard error of the proportion, we can use the formula:
Standard Error = sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
In this case, the sample proportion is 0.625 and the sample size is 400. Plugging in these values into the formula:
Standard Error = sqrt((0.625 * (1 - 0.625)) / 400)
Simplifying:
Standard Error = sqrt((0.625 * 0.375) / 400)
Standard Error = sqrt(0.234375 / 400)
Standard Error = sqrt(0.0005859375)
Standard Error ≈ 0.0241888
So, the standard error of the proportion is approximately 0.0241888.
C) To develop a 99% confidence interval for the population proportion, we can use the formula:
Confidence Interval = Sample Proportion +/- (Z * Standard Error)
Where:
- Sample Proportion is the ratio of viewers who indicated they would watch the new show and suggested it replace the crime investigation show (0.625).
- Z is the Z-score associated with the desired confidence level. For a 99% confidence level, the Z-score is approximately 2.576.
- Standard Error is the computed standard error of the proportion (0.0241888).
Plugging in the values into the formula, we have:
Confidence Interval = 0.625 +/- (2.576 * 0.0241888)
Confidence Interval ≈ 0.625 +/- 0.0623037
Therefore, the 99% confidence interval for the population proportion is approximately (0.5627, 0.6873).
Interpretation: Based on the data from the sample of 400 viewers, we can estimate with 99% confidence that the proportion of viewers who would watch the new show and suggest it replaces the crime investigation show falls between 0.5627 and 0.6873.