A college admissions officer for an MBA program has...

A college admissions officer for an MBA program has determined that historically applicants have undergraduate grade point averages that are normally distributed with standard deviation $0.45 .$ From a random sample of 25 applications from the current year, the sample mean grade point average is $2.90 .$ a. Find a $95 \%$ confidence interval for the population mean. b. Based on these sample results, a statistician computes for the population mean a confidence interval extending from $2.81$ to $2.99$. Find the confidence, level associated with this interval.

A college admissions officer for an MBA program has determined that historically applicants have undergraduate grade point averages that are normally distributed with standard deviation $0.45 .$ From a random sample of 25 applications from the current year, the sample mean grade point average is $2.90 .$
a. Find a $95 \%$ confidence interval for the population mean.
b. Based on these sample results, a statistician computes for the population mean a confidence interval extending from $2.81$ to $2.99$. Find the confidence, level associated with this interval.

Key Concepts

Confidence Level

The confidence level represents the proportion of all possible samples that can be expected to include the true population parameter. It is typically expressed as a percentage, such as 95%, meaning that if many samples were drawn and a confidence interval computed from each, about 95% of these intervals would be expected to contain the true mean. In adjustments or recalculations, different confidence levels reflect different probabilities and consequently alter the critical values used in constructing the interval.

Critical Value

A critical value is a point on the distribution (such as the z-distribution for normally distributed data) that is used to define the boundaries of the confidence interval. These values correspond to the desired confidence level and indicate how many standard errors away from the mean the interval should extend to achieve the specified level of confidence in capturing the true parameter.

Standard Error

The standard error is the standard deviation of the sampling distribution of a statistic, most commonly the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size. This measure quantifies the variability expected between the sample mean and the true population mean and is a key component in determining the margin of error for confidence intervals.

Normal Distribution

The normal distribution is a continuous probability distribution that is symmetric about its mean, where most observations cluster around the center and probabilities for values further away from the mean taper off in both directions. It is crucial in statistics because of the Central Limit Theorem, which implies that sample means are approximately normally distributed even if the underlying population distribution is not, especially when the sample size is sufficiently large. In problems where the population standard deviation is known or assumed, the normal distribution (and corresponding z-values) is used to construct confidence intervals for the population mean.

Confidence Interval

A confidence interval is a range of values, derived from sample data, that is likely to contain the true population parameter with a specified probability. In the context of mean estimation, it provides a lower and upper bound within which we expect the true mean to fall over repeated sampling, accounting for the uncertainty inherent in using a sample to estimate a population characteristic.

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