Construct a 90% confidence interval for the population mean. Assume the population has a normal distribution. A same of 15 randomly selected math majors has a grade point average of 2.86 with a standard deviation of 0.78. Round to the nearest hundredth.
Answers:
a) (2.41,3.42)
b) (2.28, 3.66)
c) (2.51, 3.21)
d) (2.37, 3.56)
CI=2.86 ± 1.76*0.78/√15
=2.86 ± 0.3545
c) (2.51, 3.21)
To construct a 90% confidence interval for the population mean, we can use the formula:
Confidence interval = sample mean ± (critical value * standard deviation / square root of sample size)
Given that we have a sample size of 15, a sample mean of 2.86, and a standard deviation of 0.78, we need to find the critical value for a 90% confidence level.
Since the population has a normal distribution, we can refer to the standard normal distribution table or use a calculator to find the critical value for a 90% confidence level.
The critical value for a 90% confidence level is approximately 1.645.
Plugging in the values into the formula, we have:
Confidence interval = 2.86 ± (1.645 * 0.78 / √15)
Calculating this expression:
Confidence interval = 2.86 ± (1.645 * 0.78 / 3.87)
Confidence interval ≈ 2.86 ± (1.127 / 3.87)
Confidence interval ≈ 2.86 ± 0.291
Therefore, the 90% confidence interval for the population mean is approximately (2.57, 3.15).
None of the given answer choices exactly match this interval.
To construct a confidence interval for the population mean, we need to follow these steps:
Step 1: Identify the sample mean and sample size.
In this case, the sample mean is 2.86 and the sample size is 15.
Step 2: Determine the standard deviation.
The standard deviation given is 0.78.
Step 3: Choose the confidence level.
The question states a 90% confidence interval.
Step 4: Calculate the margin of error.
The margin of error can be calculated using the formula: Margin of Error = (Critical Value) * (Standard Deviation / √Sample Size).
To find the critical value, we can refer to the Z-table for a 90% confidence level. The critical value is 1.645.
Substituting the values, we get Margin of Error = 1.645 * (0.78 / √15) ≈ 0.43.
Step 5: Calculate the lower and upper limits of the confidence interval.
The lower limit is the sample mean minus the margin of error, and the upper limit is the sample mean plus the margin of error.
Lower limit = 2.86 - 0.43 ≈ 2.43
Upper limit = 2.86 + 0.43 ≈ 3.29
Rounding these values to the nearest hundredth, the 90% confidence interval for the population mean is approximately (2.43, 3.29).
Comparing the possible answers given:
a) (2.41, 3.42) - This is close, but not exact.
b) (2.28, 3.66) - This interval is too wide.
c) (2.51, 3.21) - This interval is too narrow.
d) (2.37, 3.56) - This is close, but not exact.
Therefore, the correct answer is (2.43, 3.29), which is not among the provided choices.
90% = mean ± Z (SEm)
SEm = SD/√n
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (p = ±.05) and its Z score. Insert data into first equation.