The mean television viewing time for Americans is 15 hours...

The mean television viewing time for Americans is 15 hours per week (Money, November 2003 ). Suppose a sample of 60 Americans is taken to further investigate viewing habits. Assume the population standard deviation for weekly viewing time is $\sigma=4$ hours. a. What is the probability that the sample mean will be within 1 hour of the population mean? b. What is the probability that the sample mean will be within 45 minutes of the population mean?

The mean television viewing time for Americans is 15 hours per week (Money, November 2003 ). Suppose a sample of 60 Americans is taken to further investigate viewing habits. Assume the population standard deviation for weekly viewing time is $\sigma=4$ hours.
a. What is the probability that the sample mean will be within 1 hour of the population mean?
b. What is the probability that the sample mean will be within 45 minutes of the population mean?

Key Concepts

Normal Distribution Probabilities

This concept involves using the normal distribution to calculate the likelihood that a random variable falls within a certain range. By converting a sample mean into a Z-score, one can use the standard normal cumulative distribution function to compute the probability of the sample mean deviating from the population mean by specified amounts.

Z-scores

A Z-score measures how many standard deviations an element is from the mean. When applied to sample means, Z-scores allow us to convert the deviation of a sample mean from the population mean into a standardized value, enabling the use of the standard normal distribution to find probabilities.

Sampling Distribution of the Sample Mean

This concept refers to the probability distribution of all possible sample means from a given population. It centers on the true population mean and its variability decreases as the sample size increases. Understanding this distribution is key in making inferences about the population from a sample.

Central Limit Theorem

The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean will tend to be normally distributed, regardless of the shape of the population distribution, provided the sample size is sufficiently large. This concept is crucial because it allows statisticians to use normal probability methods even when the original data does not follow a normal distribution.

Standard Error

The standard error is the standard deviation of the sampling distribution of the sample mean. It is calculated by dividing the population standard deviation by the square root of the sample size. This measure indicates the accuracy with which a sample represents the population and is essential for determining probabilities related to sample means.

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