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Chapter 6: Problem 40

The average winter daily temperature in Chicago has a distribution that is approximately Normal, with a mean of 28 degrees and a standard deviation of 8 degrees. What percentage of winter days in Chicago have a daily temperature of 35 degrees or warmer? (Source: wunderground.com)

Short Answer

Expert verified
Approximately 19.1% of winter days in Chicago have a daily temperature of 35 degrees or warmer.

Step by step solution

01

Standardize the Temperature Value

First, convert the temperature value of 35 to a z-score. The z-score is a measure of how many standard deviations an element is from the mean. It can be calculated using the formula \( z = \frac{X - 渭}{蟽} \), where X is the data point, 渭 is the mean and 蟽 is the standard deviation. In this case, X is 35 (the temperature we are interested in), 渭 is 28 (the average temperature) and 蟽 is 8 (the standard deviation). This gives \( z = \frac{35 - 28}{8} \) which simplifies to \( z 鈮� 0.875 \).
02

Calculate the Area to the Right of the Z-Score

Now we need to calculate the percentage of data that lies to the right of the calculated z-score. This requires checking a standard normal distribution table or using software or a calculator with statistical capabilities. To find this value, we look for the z-score in the table which gives us the area to the left of the z-score. But we need the area to the right which is \( 1 - (Area to the left) \). If you look up the area to the left of 0.875 from the table, it will be about 0.809. We subtract this from 1 to get the area to the right. This leaves \( 1 - 0.809 = 0.191 \).
03

Convert Fraction to Percentage

The last step is to convert this decimal to a percentage, as the question asks for a percentage. Multiplying by 100, we get \( 0.191 * 100 = 19.1\% \).

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Most popular questions from this chapter

Assume that the lengths of pregnancy for humans is approximately Normally distributed, with a mean of 267 days and a standard deviation of 10 days. Use the Empirical Rule to answer the following questions. Do not use the technology or the Normal table. Begin by labeling the horizontal axis of the graph with lengths, using the given mean and standard deviation. Three of the entries are done for you. a. Roughly what percentage of pregnancies last more than 267 days? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) b. Roughly what percentage of pregnancies last between 267 and 277 days? i. \(34 \%\) iii. \(2.5 \%\) ii. \(17 \%\) iv. \(50 \%\) c. Roughly what percentage of pregnancies last less than 237 days? i. almost all iii. \(34 \%\) ii. \(50 \%\) iv. about \(0 \%\) d. Roughly what percentage of pregnancies last between 247 and 287 days? i. almost all iii. \(68 \%\) ii. \(95 \%\) iv. \(50 \%\) e. Roughly what percentage of pregnancies last longer than 287 days? i. \(34 \%\) iii. \(2.5 \%\) ii. \(17 \%\) iv. \(50 \%\) f. Roughly what percentage of pregnancies last longer than 297 days? i. almost all iii. \(34 \%\) ii. \(50 \%\) iv. about \(0 \%\)

According to the Centers of Disease Control and Prevention, \(44 \%\) of U.S. households still had landline phone service. Suppose a random sample of 60 U.S. households is taken. a. Find the probability that exactly 25 of the households sampled still have a landline. b. Find the probability that more than 25 households still have a landline. c. Find the probability that at least 25 households still have a landline. d. Find the probability that between 20 and 25 households still have a landline.

Determine whether each of the following variables would best be modeled as continuous or discrete. a. The weight of a car in pounds b. The weight of a car in kilograms

A married couple plans to have four children, and they are wondering how many boys they should expect to have. Assume none of the children will be twins or other multiple births. Also assume the probability that a child will be a boy is \(0.50 .\) Explain why this is a binomial experiment. Check all four required conditions.

Use the table or technology to find the answer to each question. Include an appropriately labeled sketch of the Normal curve for each part. Shade the appropriate region. A section of the Normal table is provided in the previous exercise. a. Find the area to the left of a \(z\) -score of \(0.92\). b. Find the area to the right of a z-score of \(0.92\).
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