Suppose the average client charge per hour for out-of-court...

Suppose the average client charge per hour for out-of-court work by lawyers in the state of Iowa is $\$ 125 .$ Suppose further that a random telephone sample of 32 lawyers in Iowa is taken and that the sample average charge per hour for out-of-court work is $\$ 110 .$ If the population variance is $\$ 525$, what is the probability of getting a sample mean of $\$ 110$ or larger? What is the probability of getting a sample mean larger than $\$ 135$ per hour? What is the probability of getting a sample mean of between $\$ 120$ and $\$ 130$ per hour?

Key Concepts

Sampling Distribution

This concept refers to the probability distribution of a statistic—in this case, the sample mean—computed from a random sample drawn from a population. Rather than considering individual observations, the sampling distribution describes how the sample means vary from sample to sample, and it forms the basis for making inferences about the population mean.

Central Limit Theorem

The Central Limit Theorem states that, regardless of the population’s distribution, the distribution of the sample means will tend to be approximately normal as the sample size increases, provided the samples are independent and identically distributed. This theorem allows us to use the normal model to calculate probabilities and make inferences about the population mean even when the population distribution is unknown.

Standard Error

The standard error is a measure of the variability of the sample mean as an estimator of the population mean. It is calculated as the population standard deviation divided by the square root of the sample size. The standard error quantifies how much the sample mean is expected to fluctuate from one sample to another, and it is essential for converting differences from the hypothesized mean into standardized units.

Z-Scores and Probability Calculations

Z-scores convert observed values into the number of standard errors they are away from the population mean, allowing us to use the standard normal distribution to calculate probabilities. By transforming sample means into z-scores, we can determine the likelihood of observing a sample mean at or beyond a specified value, which is key to hypothesis testing and confidence interval estimation.

Transcript

Please give Ace some feedback