sample of 49 observations is taken from a normal population with a standard deviation of 10. The sample mean is 55. Determine the 99 percent confidence interval for the population mean. Give the upper limit.

Question

sample of 49 observations is taken from a normal population with a standard deviation of 10. The sample mean is 55. Determine the 99 percent confidence interval for the population mean. Give the upper limit.

Answer 1

To find the 99 percent confidence interval for the population mean, we need to use the formula for the confidence interval:

Confidence interval = sample mean ± (z-value * standard deviation / square root of sample size)

Since the sample size is 49 and the standard deviation is 10, we can substitute these values into the formula:

Confidence interval = 55 ± (z-value * 10 / √49)

To find the z-value corresponding to a 99 percent confidence level, we need to find the z-value that leaves 0.5 percent in each tail of the standard normal distribution. Since the confidence level is 99 percent, the remaining 1 percent is divided equally between the two tails, giving us 0.5 percent in each tail. Looking up the z-value for 0.5 percent in a standard normal distribution table, we find that it is approximately 2.576.

Substituting these values into the formula, we get:

Confidence interval = 55 ± (2.576 * 10 / √49)

Simplifying this equation:

Confidence interval = 55 ± (25.76 / 7)

Confidence interval = 55 ± 3.68

Therefore, the upper limit of the 99 percent confidence interval for the population mean is 55 + 3.68 = 58.68.

Answer 2

To determine the 99 percent confidence interval for the population mean, we can use the formula:

Confidence Interval = sample mean ± (z * standard deviation / square root of sample size)

First, we need to find the value of 'z'. Since we want a 99 percent confidence interval, we need to find the z-value corresponding to a 0.995 confidence level. This can be done using a z-table or calculator. The z-value corresponding to a 0.995 confidence level is approximately 2.576.

Next, let's calculate the confidence interval:

Confidence Interval = 55 ± (2.576 * 10 / sqrt(49))

Calculating the square root of 49 gives us 7, so:

Confidence Interval = 55 ± (2.576 * 10 / 7)

Now, let's calculate the upper limit of the confidence interval:

Upper Limit = 55 + (2.576 * 10 / 7)

Solving this equation, we get:

Upper Limit = 57.712

Therefore, the upper limit of the 99 percent confidence interval for the population mean is 57.712.

Answer 3

To determine the 99 percent confidence interval for the population mean, you can use the formula:

Confidence Interval = sample mean ± (Z * standard deviation / √n)

where:
- sample mean = 55
- standard deviation = 10
- n = sample size (49 in this case)
- Z = Z-score, which corresponds to the desired confidence level

First, find the Z-score that corresponds to a 99 percent confidence level. You can use a standard normal distribution table or a calculator to find this value. For a 99 percent confidence level, the Z-score is approximately 2.576.

Now, substitute the values into the formula:

Confidence Interval = 55 ± (2.576 * 10 / √49)

Simplifying this expression gives us:

Confidence Interval = 55 ± (2.576 * 10 / 7)

Calculating the values:

Confidence Interval = 55 ± 3.696

To find the upper limit, add the result to the sample mean:

Upper Limit = 55 + 3.696 = 58.696

Therefore, the upper limit of the 99 percent confidence interval for the population mean is approximately 58.696.