Week 4 - Assignment-Heather Montoya z score formula

Student: Heather Montoya Date: 10/21/23 Instructor: Jean Insinga Course: MAT232: Statistical Literacy (GSR2337B) Assignment: Week 4 - Assignment Suppose certain coins have weights that are normally distributed with a mean of and a standard deviation of . A vending machine is configured to accept those coins with weights between g and g. 5.940 g 0.075 g 5.820 6.060 Click here to view page 1 of the standard normal distribution. 1 Click here to view page 2 of the standard normal distribution. 2 a. If different coins are inserted into the vending machine, what is the expected number of rejected quarters? 250 The expected number of rejections is based on the probability for each test of a coin. Each test is an individual value from a normally distributed population. Therefore, use the population distribution to determine the probability. First, convert the given weights to their corresponding z scores using z and the population distribution statistics. = x − z = 5.820 − 5.940 0.075 − 1.6 z = 6.060 − 5.940 0.075 1.6 The probability is the area to the left of and the area to the right of under the standard normal distribution. z = − 1.6 z = 1.6 First find the area to the left of using a table of the standard normal distribution z = − 1.6 The area to the left of is . z = − 1.6 0.0548 Notice that the right z score is the opposite of the left z score. Since the normal distribution is symmetric, the area to the right of is the same as the area to the left of . Add to find the total area. z = 1.6 z = − 1.6 0.0548 + 0.0548 = 0.1096 Therefore, the probability that a randomly selected coin's weight is rejected is approximately . Multiply the probability by the number of tests to determine the number of expected rejections. 0.1096 0.1096 • 250 = 27 b. If different coins are inserted into the vending machine, what is the probability that the mean falls between the limits of g and g? 250 5.820 6.060 In this case, the desired probability is for the mean of a sample of coins. Therefore, use the central limit theorem. 250 According to the central limit theorem, the distribution of sample means will have a mean given by and a standard deviation given by . x = x = x n The mean of the distribution of sample means is the same as the population mean, so . x x = 5.940 Apply the definition for the standard deviation of the distribution of the sample means for a sample size of . 250 x = n = 0.075 250 Simplify to find the standard deviation of the distribution of the sample means. x = 0.075 250 0.004743 Therefore, the distribution of sample means for a sample size of is approximately normal with a mean and a standard deviation Use these values to compute the corresponding z score for each height. x 250 = 5.940 x = 0.004743. x z 5.820 − 5.940 0.004743 − 25.3 z 6.060 − 5.940 0.004743 25.3 The probability is the area between and under the standard normal distribution. z = − 25.3 z = 25.3 Find the area by subtracting the area to the left of from the area to the left of . Use a table to find each of these areas. z = − 25.3 z = 25.3 = 5.940 x = 5.820 x = 6.060 x = 5.940 x = 5.820 x = 6.060 x

1: Standard Normal Distribution Table (Page 1) 2: Standard Normal Distribution Table (Page 2) The area to the left of is . z = − 25.3 0.0001 The area to the left of is . z = 25.3 0.9999 Subtract to find the area between z and z . = − 25.3 = 25.3 0.9999 − 0.0001 = 0.9998 Therefore, the probability that the mean weight of a sample of randomly selected coins is between g and g is approximately . 250 5.820 6.060 0.9998 c. Which results is more important to the owner of the vending machine? Why? If the number of rejected coins is significant, then the result from part (a) is more important. If the average weight of the coins is significant, then the result from part (b) is more important.