MATH1127: Introduction to Statistics (60022) Unit 3 Test Practice Quizzes Quiz: Chapter 7
Quiz: Chapter 7
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Question 1
A manufacturer knows that their items have a lengths that are approximately normally distributed, with a mean of 12.3 inches, and standard deviation of 1 inches.

If 39 items are chosen at random, what is the probability that their mean length is greater than 12 inches?
(Round answer to four decimal places)

Step 1 ：The problem is asking for the probability that the mean length of 39 randomly chosen items is greater than 12 inches. This is a problem of normal distribution. The mean of the distribution is given as 12.3 inches and the standard deviation is 1 inch.

Step 2 ：We can use the Central Limit Theorem which states that the distribution of sample means approximates a normal distribution as the sample size gets larger, regardless of the shape of the population distribution.

Step 3 ：The standard deviation of the sample mean (also known as the standard error) can be calculated as the standard deviation of the population divided by the square root of the sample size. In this case, the standard error is \(\frac{1}{\sqrt{39}}\) which is approximately 0.16012815380508713.

Step 4 ：We can then calculate the z-score, which measures how many standard deviations an element is from the mean. The z-score for a value x is calculated as \(\frac{x - \text{mean}}{\text{standard error}}\). In this case, the z-score is \(\frac{12 - 12.3}{0.16012815380508713}\) which is approximately -1.873499399519524.

Step 5 ：Finally, we can use the z-score to find the probability that the mean length is greater than 12 inches. This can be done by finding the area under the normal distribution curve to the right of the z-score, which represents the probability. The probability is approximately 0.9695.

Step 6 ：Final Answer: The probability that the mean length of 39 randomly chosen items is greater than 12 inches is \(\boxed{0.9695}\).