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Chapter 7: Problem 45

Let \(x\) be a continuous random variable that has a normal distribution with \(\mu=48\) and \(\sigma=8\). Assuming \(n / N \leq .05\), find the probability that the sample mean, \(\bar{x}\), for a random sample of 16 taken from this population will be a. between \(49.6\) and \(52.2\) b. more than \(45.7\)

Short Answer

Expert verified
Since we're dealing with a standard normal distribution, the Z-table or standard normal table is used to find the probabilities. In part a, the probability that the sample mean is between \(49.6\) and \(52.2\) is \(P(z_1 \leq z \leq z_2) = P(z \leq z_2) - P(z \leq z_1)\). In part b, the probability that the sample mean is more than \(45.7\) is \(P(z \geq z_3) = 1 - P(z \leq z_3)\). The exact probabilities depend on the calculated z-scores and the values in the Z-table.

Step by step solution

01

Identify given parameters

In this problem, we have a normally distributed variable \(x\) with a mean \(\mu = 48\) and a standard deviation \(\sigma = 8\). The size of the sample drawn is \(n=16\).
02

Convert to Standard Normal Distribution (Z-Distribution) for Part a

In part a, we are looking for the probability that the sample mean will be between \(49.6\) and \(52.2\). We find the corresponding z-scores using the z-score formula: \(z = (\bar{x} - \mu) / (\sigma / \sqrt{n})\). The z-scores for \(49.6\) and \(52.2\) are \(z_1= (\bar{x_1} - \mu) / (\sigma / \sqrt{n})\) and \(z_2 = (\bar{x_2} - \mu) / (\sigma / \sqrt{n})\), respectively.
03

Find Probability for Part a

Use the standard normal distribution table to find the probabilities that correspond to \(z_1\) and \(z_2\). The required probability is then \(P(z_1 \leq z \leq z_2) = P(z \leq z_2) - P(z \leq z_1)\).
04

Convert to Standard Normal Distribution (Z-Distribution) for Part b

Similarly for part b, we need to find the probability that the sample mean will be more than \(45.7\). Calculate the z-score \(z_3\) for \(\bar{x_3} = 45.7\).
05

Find Probability for Part b

Use the standard normal distribution table to find the probability corresponding to \(z_3\). The required probability is given by \(P(z \geq z_3) = 1 - P(z \leq z_3)\).

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Most popular questions from this chapter

If all possible samples of the same (large) size are selected from a population, what percentage of all the sample means will be within \(2.5\) standard deviations of the population mean?

As mentioned in Exercise \(7.22\), the average cost of going to a minor league baseball game for a family of four was \(\$ 55\) in 2009 . Suppose that the standard deviation of such costs is \(\$ 13.25 .\) Find the probability that the average cost of going to a minor league baseball game for 33 randomly selected such families is a. more than \(\$ 60\) b. less than \(\$ 52\) c. \(\$ 54\) to \(\$ 57.99\)

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