a. A random sample of 25 was drawn from a normal distribution with a standard deviation of 5. The sample mean is 80. Determine the 95% confidence interval estimate of the population mean.
Formula:
CI95 = mean + or - 1.96(sd/√n)
sd = standard deviation
n = sample size
Substitute the values from your problem into the formula and calculate.
To determine the 95% confidence interval estimate of the population mean, we need to use the following formula:
Confidence Interval = sample mean +/- margin of error
The margin of error is calculated using the formula:
Margin of Error = (critical value) * (standard deviation / square root of sample size)
Step 1: Find the critical value associated with a 95% confidence level. This critical value corresponds to the z-score for a confidence level of 95%. We can use a standard normal distribution table or a statistical calculator to find the critical value.
For a 95% confidence level, the critical value is approximately 1.96.
Step 2: Calculate the margin of error.
Margin of Error = 1.96 * (5 / sqrt(25))
= 1.96 * (5 / 5)
= 1.96 * 1
= 1.96
Step 3: Finally, calculate the confidence interval.
Confidence Interval = 80 +/- 1.96
= (80 - 1.96, 80 + 1.96)
= (78.04, 81.96)
The 95% confidence interval estimate of the population mean is (78.04, 81.96).
To determine the 95% confidence interval estimate of the population mean, you need to use the formula:
Confidence interval = sample mean ± (critical value) * (standard deviation / square root of sample size)
Step 1: Calculate the critical value. For a 95% confidence level, you can use a Z-table or a Z-value calculator to find the critical value. The Z-value for a 95% confidence level is 1.96.
Step 2: Plug in the values into the formula.
Sample mean = 80
Standard deviation = 5
Square root of sample size = square root of 25 = 5 (since 25 is a perfect square)
Confidence interval = 80 ± (1.96) * (5 / 5)
Simplifying the equation, you get:
Confidence interval = 80 ± 1.96
Step 3: Calculate the lower and upper bounds of the confidence interval.
Lower bound = 80 - 1.96 = 78.04
Upper bound = 80 + 1.96 = 81.96
Therefore, the 95% confidence interval estimate of the population mean is (78.04, 81.96). This means we are 95% confident that the true population mean falls within this range.