The undergraduate grade point (GPA) for students admitted to the top graduate business shools was 3.37 assume this estimate was based on a sample of 120 students admited to the top schools, using past year data the population standard deviation can be assumed known with standard deviation = .28 what is the 95% confidence interval estimate of the mean undergraduate GPA for students admited to the top graduate business school?
We can use the formula:
( X̄ - z*(σ/√n), X̄ + z*(σ/√n) )
where:
X̄ = sample mean = 3.37
z = z-score for desired confidence level = 1.96 (from a standard normal distribution)
σ = population standard deviation = 0.28
n = sample size = 120
Plugging in the values, we get:
( 3.37 - 1.96*(0.28/√120), 3.37 + 1.96*(0.28/√120) )
Simplifying, we get:
( 3.28, 3.46 )
Therefore, we are 95% confident that the true mean undergraduate GPA for students admitted to the top graduate business schools is between 3.28 and 3.46.
To calculate the 95% confidence interval estimate of the mean undergraduate GPA for students admitted to the top graduate business school, we can use the formula:
Confidence Interval = sample mean ± (critical value * standard error)
Where:
- Sample mean: the mean GPA of the sample (3.37)
- Critical value: the value obtained from the standard normal distribution corresponding to the desired confidence level (95% confidence interval corresponds to a critical value of 1.96)
- Standard error: the standard deviation of the population divided by the square root of the sample size (0.28 / √120)
Let's calculate the confidence interval step by step:
1. Calculate the standard error:
Standard error = 0.28 / √120 ≈ 0.0255
2. Calculate the margin of error:
Margin of error = Critical value * Standard error
Margin of error = 1.96 * 0.0255 ≈ 0.05
3. Calculate the lower and upper bounds of the confidence interval:
Lower bound = Sample mean - Margin of error
Lower bound = 3.37 - 0.05 ≈ 3.32
Upper bound = Sample mean + Margin of error
Upper bound = 3.37 + 0.05 ≈ 3.42
Therefore, the 95% confidence interval estimate of the mean undergraduate GPA for students admitted to the top graduate business school is approximately 3.32 to 3.42.