A 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 lower limit.
To calculate the confidence interval for the population mean, we will use the formula:
Confidence Interval = sample mean ± (critical value * standard deviation / √sample size)
The critical value can be found using the z-table for a 99% confidence level. Since we want to find the lower limit, we will subtract the margin of error from the sample mean.
First, let's find the critical value:
The 99% confidence level will leave 1% for each tail, so we divide it by 2 to get 0.5% in each tail.
Looking up the z-value in the z-table for 0.5% (or 0.005), we find a value of -2.576.
Now, we can calculate the confidence interval:
Confidence Interval = 55 ± (-2.576 * 10 / √49)
Confidence Interval = 55 ± (-2.576 * 10 / 7)
Confidence Interval = 55 ± (-3.668)
To find the lower limit, we subtract the margin of error from the sample mean:
Lower Limit = 55 - 3.668
Lower Limit = 51.332
Therefore, the lower limit of the 99% confidence interval for the population mean is 51.332.
To determine the 99 percent confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean ± (Z * (Standard Deviation / √(Sample Size)))
First, let's calculate the value of Z for a 99 percent confidence level. The confidence level is the complement of the significance level, which gives us α = 1 - 0.99 = 0.01. Since the values are normally distributed, we can use a standard normal distribution table or a calculator to find the corresponding Z-score.
Looking up the Z-score for a 0.01 probability in a standard normal distribution table, we find the Z-score to be approximately -2.576.
Now, we can calculate the confidence interval using the given information:
Confidence Interval = 55 ± (-2.576 * (10 / √(49)))
Calculating the standard error first:
Standard Error = Standard Deviation / √(Sample Size)
= 10 / √(49)
= 10 / 7
= 1.43 (rounded to two decimal places)
Now, substitute the values into the confidence interval formula:
Confidence Interval = 55 ± (-2.576 * 1.43)
Calculating the lower limit:
Lower Limit = Sample Mean - (Z * Standard Error)
= 55 - (-2.576 * 1.43)
= 55 + 3.678
= 58.68 (rounded to two decimal places)
Therefore, the lower limit of the 99 percent confidence interval for the population mean is approximately 58.68.
To determine the 99 percent confidence interval for the population mean, you need to use the formula:
Confidence interval = sample mean ± (z-value) * (standard deviation / √n)
Where:
- Sample mean is the mean of the sample, which is given as 55 in this case.
- z-value represents the critical value for the desired confidence level. For a 99 percent confidence level, the z-value is approximately 2.576.
- Standard deviation is the standard deviation of the population, given as 10 here.
- n is the number of observations in the sample, which is also given as 49.
Now, let's calculate the confidence interval:
Confidence interval = 55 ± (2.576) * (10 / √49)
First, find the value of √49, which is 7.
Confidence interval = 55 ± (2.576) * (10 / 7)
Next, calculate the value inside the parentheses: (2.576) * (10 / 7) ≈ 3.697
Finally, calculate the confidence interval:
Confidence interval = 55 ± 3.697
To find the lower limit of the confidence interval, subtract the value calculated from the sample mean:
Lower limit = 55 - 3.697
Lower limit ≈ 51.303
Therefore, the lower limit of the 99 percent confidence interval for the population mean is approximately 51.303.