A simple random sample of 50 items from a population with σ � 6 resulted in a sample
mean of 32.
a. Provide a 90% confidence interval for the population mean
90% = mean ± 1.645 SEm
SEm = SD/√n
1.396
To provide a 90% confidence interval for the population mean, we can use the formula:
Confidence Interval = Sample Mean ± (Z * Standard Error)
Where:
- Sample Mean = 32 (given in the problem)
- Z = Z-score for a 90% confidence interval (we can find this value in the standard normal distribution table or use a calculator)
- Standard Error = σ / √n (standard deviation divided by the square root of the sample size)
Since the population standard deviation (σ) is given as 6 and the sample size (n) is 50, we can calculate the standard error:
Standard Error = 6 / √50
Now, we need to determine the Z-score for a 90% confidence interval. The Z-score value associated with a 90% confidence interval is approximately 1.645.
Plugging the values into the formula, we get:
Confidence Interval = 32 ± (1.645 * (6 / √50))
Now, let's calculate the confidence interval step-by-step:
First, calculate the standard error:
Standard Error = 6 / √50 ≈ 0.8485 (rounded to four decimal places)
Next, multiply the Z-score by the standard error:
1.645 * 0.8485 ≈ 1.3979 (rounded to four decimal places)
Finally, calculate the confidence interval:
Lower Limit = 32 - 1.3979 ≈ 30.60 (rounded to two decimal places)
Upper Limit = 32 + 1.3979 ≈ 33.40 (rounded to two decimal places)
Therefore, the 90% confidence interval for the population mean is approximately 30.60 to 33.40.
To calculate a 90% confidence interval for the population mean, we'll use the formula:
Confidence interval = sample mean ± Margin of Error
First, let's calculate the margin of error.
The margin of error is given by the formula:
Margin of Error = Z * (σ/√n)
Where:
Z is the z-score corresponding to the desired confidence level (90%), which can be found using a standard normal distribution table or calculator.
σ is the population standard deviation. In this case, it is given as 6.
n is the sample size. In this case, it is 50.
Since we're looking for a 90% confidence interval, we need to find the z-score corresponding to a confidence level of 95%, which is 1.645.
Now, let's substitute the values in the formula:
Margin of Error = 1.645 * (6/√50)
Calculating this, we have:
Margin of Error ≈ 1.645 * (6/√50) ≈ 1.645 * 0.848 ≈ 1.39676
Next, we can calculate the confidence interval by adding and subtracting the margin of error from the sample mean:
Confidence interval = 32 ± 1.39676
Calculating this, we have:
Lower bound = 32 - 1.39676 ≈ 30.60324
Upper bound = 32 + 1.39676 ≈ 33.39676
Therefore, the 90% confidence interval for the population mean is approximately (30.60, 33.40).