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.
D) Interpret your findings.
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 / sqrt(sample size))
First, let's find the critical value for a 99% confidence level. Since the sample size is large (n > 30), we can use the z-distribution. The critical value for a 99% confidence level is approximately 2.576.
Confidence Interval = 55 ± (2.576 * 10 / sqrt(49))
Confidence Interval = 55 ± 14.12
Therefore, the 99% confidence interval for the population mean is (40.88, 69.12).
2)
A) To estimate the value of the population proportion, we divide the number of viewers who indicated they would watch the new show by the sample size.
Population Proportion = Number of viewers who indicated watching the new show / Sample size
Population Proportion = 250 / 400
Population Proportion = 0.625 or 62.5%
B) To compute the standard error of the proportion, we use the formula:
Standard Error of Proportion = sqrt((p * (1-p)) / n)
Where p is the estimated population proportion and n is the sample size.
Standard Error of Proportion = sqrt((0.625 * (1-0.625)) / 400)
Standard Error of Proportion = sqrt(0.234375 / 400)
Standard Error of Proportion = 0.0305
C) To develop a 99% confidence interval for the population proportion, we can use the formula:
Confidence Interval = Sample proportion ± (critical value * Standard Error of Proportion)
Since the sample size is large (n > 30), we can use the z-distribution. The critical value for a 99% confidence level is approximately 2.576.
Confidence Interval = 0.625 ± (2.576 * 0.0305)
Confidence Interval = 0.625 ± 0.0785
Therefore, the 99% confidence interval for the population proportion is (0.5465, 0.7035) or (54.65%, 70.35%).
D) This means that we can be 99% confident that the true population proportion of viewers who would watch the new show and suggest replacing the crime investigation show lies between 54.65% and 70.35%.
1) To determine the 99% confidence interval for the population mean, we can use the formula:
Confidence Interval = sample mean ± (Z * (sample standard deviation / √sample size))
First, let's find the Z-score for a 99% confidence level. You can use a Z-table or a statistical software to find this value. The Z-score for a 99% confidence level is approximately 2.576.
Next, plug the values into the formula:
Confidence Interval = 55 ± (2.576 * (10 / √49))
Simplifying the equation:
Confidence Interval = 55 ± (2.576 * (10 / 7))
Calculating the values:
Confidence Interval = 55 ± (2.576 * 1.4286)
Confidence Interval = 55 ± 3.6864
The 99% confidence interval for the population mean is (51.3136, 58.6864).
2) A) To estimate the value of the population proportion, we can use the formula:
Population Proportion = (number of successes) / (sample size)
In this case, the number of successes (people indicating they would watch the new show and suggesting it replace the crime investigation show) is 250, and the sample size is 400.
Population Proportion = 250 / 400
Population Proportion = 0.625
So, the estimated value of the population proportion is 0.625, or 62.5%.
B) To compute the standard error of the proportion, we can use the formula:
Standard Error = √((population proportion * (1 - population proportion)) / sample size)
Plugging in the values:
Standard Error = √((0.625 * (1 - 0.625)) / 400)
Standard Error = √((0.625 * 0.375) / 400)
Standard Error = √(0.234375 / 400)
Standard Error = √0.0005859375
Standard Error = 0.0241935
The standard error of the proportion is approximately 0.0242.
C) To develop a 99% confidence interval for the population proportion, we can use the formula:
Confidence Interval = sample proportion ± (Z * standard error)
For a 99% confidence level, the Z-score is approximately 2.576.
Confidence Interval = 0.625 ± (2.576 * 0.0241935)
Calculating the values:
Confidence Interval = 0.625 ± (2.576 * 0.0241935)
Confidence Interval = 0.625 ± 0.062326
The 99% confidence interval for the population proportion is approximately (0.5627, 0.6873), or 56.27% to 68.73%.
D) To interpret the findings, we can say that with 99% confidence, the true population proportion of viewers who would watch the new show and suggest it replace the crime investigation show is estimated to be between 56.27% and 68.73%.