Calculate the following:
a) Determine the sample size necessary to estimate a population mean to within 1 with a 90% confidence given that the population standard deviation is 10.
b) Suppose that the sample was calculated at 150. Estimate the population mean with 90% confidence.
Formula for a):
n = [(z-value * sd)/E]^2
...where n = sample size, z-value will be found using a z-table to represent the 90% confidence interval, sd = 10, E = 1, ^2 means squared, and * means to multiply.
Plug the values into the formula and finish the calculation. Round your answer to the next highest whole number.
For b) Use a confidence interval formula:
CI90 = mean + or - z-value (sd/√n)
I'll let you take it from here.
10
To determine the sample size necessary to estimate a population mean to within 1 with a 90% confidence, you would use the formula:
n = (Z * σ / E)^2
where:
n = sample size
Z = Z-score corresponding to the desired confidence level (in this case, 90% confidence corresponds to a Z-score of 1.645)
σ = population standard deviation
E = margin of error
In this case, the population standard deviation is given as 10 and the desired margin of error is 1, so substituting these values into the formula gives:
n = (1.645 * 10 / 1)^2
n = 267.28
Round the value of n to the nearest whole number to get the sample size necessary to estimate the population mean to within 1 with a 90% confidence, which is 268.
b) To estimate the population mean with a 90% confidence given a sample size of 150, you would calculate the confidence interval using the formula:
CI = X̄ ± Z * (σ / √n)
where:
CI = confidence interval
X̄ = sample mean
Z = Z-score corresponding to the desired confidence level (1.645 for 90% confidence)
σ = population standard deviation
n = sample size
Since the standard deviation (σ) is given as 10, the sample size (n) is given as 150, and you want to estimate the population mean, the formula becomes:
CI = X̄ ± 1.645 * (10 / √150)
Now, you need the value of X̄ (the sample mean) to complete the calculation. If you have the value of X̄, you can substitute it into the formula to find the confidence interval.
a) To determine the sample size necessary to estimate a population mean to within 1 with a 90% confidence given that the population standard deviation is 10, you can use the formula for sample size calculation:
n = (Z * σ / E)²
Where:
n = sample size
Z = z-score representing the desired level of confidence (in this case, 90% confidence corresponds to a z-score of approximately 1.645)
σ = population standard deviation
E = desired margin of error (in this case, 1)
Plugging in the values into the formula:
n = (1.645 * 10 / 1)²
n = 26.99
Rounding up to the nearest whole number, the sample size needed is 27.
b) To estimate the population mean with 90% confidence given that the sample size is 150, you can use the formula for confidence interval calculation:
Confidence Interval = (sample mean) ± (Z * σ / sqrt(n))
Where:
sample mean = 150 (given)
Z = z-score representing the desired level of confidence (in this case, 90% confidence corresponds to a z-score of approximately 1.645)
σ = population standard deviation (given as 10)
n = sample size (given as 150)
Plugging in the values into the formula:
Confidence Interval = 150 ± (1.645 * 10 / sqrt(150))
Calculating the value inside the parentheses first:
1.645 * 10 / √150 ≈ 1.34
Therefore, the 90% confidence interval for the population mean is approximately:
150 ± 1.34
This means that we estimate the population mean to be within the range of (150 - 1.34) to (150 + 1.34) with 90% confidence.