Suppose x has a normal distribution with mean μ = 35 and standard deviation σ = 9.
(a) Describe the distribution of x values for sample size n = 4. (Use 2 decimal places.)
μx =
σx =
(b) Describe the distribution of x values for sample size n = 16. (Use 2 decimal places.)
μx =
σx =
(c) Describe the distribution of x values for sample size n = 100. (Use 2 decimal places.)
μx =
σx =
Expect the µ's to stay the same.
The standard error of the mean (SEm) = SD/√n
SEm would tend to get smaller as n increases.
(a) For sample size n = 4, the distribution of x values can be described as follows:
- Mean of the sample means (μx) = 35
- Standard deviation of the sample means (σx) = σ/√n = 9/√4 = 4.50
(b) For sample size n = 16, the distribution of x values can be described as follows:
- Mean of the sample means (μx) = 35
- Standard deviation of the sample means (σx) = σ/√n = 9/√16 = 2.25
(c) For sample size n = 100, the distribution of x values can be described as follows:
- Mean of the sample means (μx) = 35
- Standard deviation of the sample means (σx) = σ/√n = 9/√100 = 0.90
To describe the distribution of x values for different sample sizes, we need to calculate the mean (μx) and standard deviation (σx) of the sample means.
(a) For sample size n = 4:
To calculate μx, we divide the population mean (μ) by the square root of the sample size:
μx = μ / √n = 35 / √4 = 35 / 2 = 17.50 (rounded to 2 decimal places)
To calculate σx, we divide the population standard deviation (σ) by the square root of the sample size:
σx = σ / √n = 9 / √4 = 9 / 2 = 4.50 (rounded to 2 decimal places)
Therefore, for sample size n = 4, the mean of the sample means (μx) is 17.50 and the standard deviation of the sample means (σx) is 4.50.
(b) For sample size n = 16:
μx = μ / √n = 35 / √16 = 35 / 4 = 8.75 (rounded to 2 decimal places)
σx = σ / √n = 9 / √16 = 9 / 4 = 2.25 (rounded to 2 decimal places)
For sample size n = 16, the mean of the sample means (μx) is 8.75 and the standard deviation of the sample means (σx) is 2.25.
(c) For sample size n = 100:
μx = μ / √n = 35 / √100 = 35 / 10 = 3.50 (rounded to 2 decimal places)
σx = σ / √n = 9 / √100 = 9 / 10 = 0.90 (rounded to 2 decimal places)
For sample size n = 100, the mean of the sample means (μx) is 3.50 and the standard deviation of the sample means (σx) is 0.90.