University Costs. After deducting grants based on need, the...

University Costs. After deducting grants based on need, the average cost to attend the University of Southern Califormia (USC) is $$\$ 27,175$$. Assume the population standard deviation is $$\$ 7400$$. Suppose that a random sample of 60 USC students will be taken from this population. a. What is the value of the standard error of the mean? b. What is the probability that the sample mean will be more than $$\$ 27,175$$ ? c. What is the probability that the sample mean will be within $$\$ 1000$$ of the population mean? d. How would the probability in part (c) change if the sample size were increased to 100 ?

University Costs. After deducting grants based on need, the average cost to attend the University of Southern Califormia (USC) is $$\$ 27,175$$. Assume the population standard deviation is $$\$ 7400$$. Suppose that a random sample of 60 USC students will be taken from this population.
a. What is the value of the standard error of the mean?
b. What is the probability that the sample mean will be more than $$\$ 27,175$$ ?
c. What is the probability that the sample mean will be within $$\$ 1000$$ of the population mean?
d. How would the probability in part (c) change if the sample size were increased to 100 ?

Key Concepts

Central Limit Theorem

The Central Limit Theorem states that, regardless of the population's distribution, the sampling distribution of the sample mean will tend to be approximately normally distributed when the sample size is sufficiently large. This theorem underpins many statistical procedures by justifying the use of normal probability models for sample means, even when the underlying distribution is not normal.

Standard Error of the Mean

The standard error of the mean measures the amount of variability in sample means around the population mean. It is calculated by dividing the population standard deviation by the square root of the sample size. A smaller standard error indicates that the sample mean is a more precise estimate of the population mean.

Z-Score and Standard Normal Distribution

A z-score represents the number of standard deviations a data point, such as a sample mean, is from the population mean. By converting a sample mean to a z-score, one can use the standard normal distribution to calculate probabilities associated with deviations from the mean, which is essential in hypothesis testing and estimation.

Sample Size and Precision

Sample size plays a critical role in determining the precision of the sample mean as an estimate of the population mean. Increasing the sample size decreases the standard error, which in turn increases the probability that the sample mean will fall within a specified range around the population mean, thereby improving the accuracy of statistical inferences.

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