The following simple random sample was selected from a normal distribution: 4, 6, 3, 5, 9, and 3.
a. Construct a 90% confidence interval for the population mean μ.
To construct a confidence interval for the population mean μ, we can use the formula:
Confidence interval = sample mean ± (critical value * standard error)
In this case, we have a simple random sample of 6 observations. Let's go step by step to calculate the confidence interval:
1. Calculate the sample mean:
Add up all the values in the sample and divide by the sample size.
Sample mean = (4 + 6 + 3 + 5 + 9 + 3) / 6 = 5
2. Calculate the standard deviation (SD):
Subtract the sample mean from each observation, square the differences, sum them up, and divide by the sample size minus one. Then take the square root.
First, calculate the squared differences:
(4 - 5)^2 + (6 - 5)^2 + (3 - 5)^2 + (5 - 5)^2 + (9 - 5)^2 + (3 - 5)^2 = 4 + 1 + 4 + 0 + 16 + 4 = 29
Now, divide by sample size minus one:
29 / (6 - 1) = 29 / 5 = 5.8
Finally, take the square root:
SD = √(5.8) ≈ 2.41
3. Calculate the critical value (z-value):
The critical value depends on the desired confidence level and the sample size. For a 90% confidence level, the critical value is 1.645 for a sample size of 6.
4. Calculate the standard error:
The standard error represents the uncertainty of the sample mean and is calculated by dividing the standard deviation by the square root of the sample size.
Standard error = SD / √(sample size) = 2.41 / √(6) ≈ 0.984
Remember to round to an appropriate number of decimal places.
5. Calculate the confidence interval:
Now we have all the values we need:
Sample mean = 5
Critical value = 1.645
Standard error = 0.984
Apply the formula:
Confidence interval = 5 ± (1.645 * 0.984)
Confidence interval ≈ 5 ± 1.615
So, the 90% confidence interval for the population mean μ is approximately (3.385, 6.615).
Note: The confidence interval provides a range of values within which we can reasonably expect the population mean to fall with a certain level of confidence.
To construct a confidence interval for the population mean (μ), we can use the formula:
Confidence Interval = sample mean ± margin of error
1. Find the sample mean (x̄):
x̄ = (4 + 6 + 3 + 5 + 9 + 3) / 6
x̄ = 30 / 6
x̄ = 5
2. Find the standard deviation (s) of the sample:
s = √[(Σ(xi - x̄)^2) / (n - 1)]
where xi is each individual value in the sample, x̄ is the sample mean, and n is the sample size.
s = √[( (4-5)^2 + (6-5)^2 + (3-5)^2 + (5-5)^2 + (9-5)^2 + (3-5)^2) / (6-1)]
s = √[( (-1)^2 + (1)^2 + (-2)^2 + (0)^2 + (4)^2 + (-2)^2) / 5]
s = √[(1 + 1 + 4 + 0 + 16 + 4) / 5]
s = √[26 / 5]
s ≈ √5.2
s ≈ 2.28
3. Calculate the standard error (SE) using the formula:
SE = s / √n
where s is the standard deviation and n is the sample size.
SE = 2.28 / √6
SE ≈ 0.93
4. Determine the margin of error (ME) using the t-distribution:
ME = critical value * SE
Since we want a 90% confidence interval, we need to find the critical value for a 95% confidence level (1 - 0.90 = 0.10).
Using a t-table or calculator, the critical value for a sample size of 6 and a confidence level of 95% is approximately 1.943.
ME = 1.943 * 0.93
ME ≈ 1.80
5. Construct the confidence interval:
Confidence Interval = x̄ ± ME
Confidence Interval = 5 ± 1.80
Confidence Interval = (3.20, 6.80)
Therefore, the 90% confidence interval for the population mean μ is (3.20, 6.80).