/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 23 Cool pool? Coach Ferguson uses a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 23

Cool pool? Coach Ferguson uses a thermometer to measure the temperature (in degrees Celsius) at 20 different locations in the school swimming pool. An analysis of the data yields a mean of \(25^{\circ} \mathrm{C}\) and a standard deviation of \(2^{\circ} \mathrm{C}\). Find the mean and standard deviation of the temperature readings in degrees Fahrenheit (recall that \({ }^{\circ} \mathrm{F}=(9 / 5)^{\circ} \mathrm{C}+32\) ).

Short Answer

Expert verified
Mean: \(77^{\circ}F\); Standard Deviation: \(3.6^{\circ}F\).

Step by step solution

01

Convert the Mean to Fahrenheit

First, we need to convert the mean temperature from Celsius to Fahrenheit using the conversion formula. The given mean is \(25^{\circ}\,C\). The formula for converting Celsius to Fahrenheit is: \(^{\circ}F = \frac{9}{5}\times^{\circ}C+32\). Thus, \(^{\circ}F = \frac{9}{5}\times 25 + 32\). Calculating this gives: \(^{\circ}F = 45 + 32 = 77^{\circ}F\). So, the mean temperature in Fahrenheit is \(77^{\circ}F\).
02

Convert the Standard Deviation to Fahrenheit

The spread or scale of the data is affected only by the multiplication part of the linear transformation (conversion equation) because adding a constant (32) does not impact the spread. Given that the standard deviation in Celsius is \(2^{\circ}\,C\), we use the formula only with the multiplication term: \( SD_{F} = \frac{9}{5} \times SD_{C}\).Thus, \( SD_{F} = \frac{9}{5} \times 2 = 3.6\).So, the standard deviation of the temperatures in Fahrenheit is \(3.6^{\circ}F\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mean Temperature Conversion
Understanding how to convert the mean temperature from Celsius to Fahrenheit requires us to use a standard conversion formula. This formula is:
  • \(^\circ F = \left(\frac{9}{5}\right) \times ^\circ C + 32\)
The conversion process has two steps. First, multiply the Celsius mean by \(\frac{9}{5}\). This scales the temperature to match the Fahrenheit system. Then, add 32 to account for zero offset between these temperature scales.
In our exercise, the mean temperature is \(25^{\circ}C\). Applying this conversion, we get:
  • \(^\circ F = \left(\frac{9}{5}\right) \times 25 + 32\)
Upon calculating, it becomes:
  • \(45 + 32 = 77^{\circ}F\)
This result shows the average swimming pool temperature in Fahrenheit, underscoring that converting a mean temperature involves both scaling and shifting the original value.
Standard Deviation Conversion
Converting the standard deviation from Celsius to Fahrenheit involves a slightly different process than converting the mean. Standard deviation measures the spread of your data, so the constant you add when converting to Fahrenheit does not affect its calculation.
Instead, we only consider the multiplication factor, which pertains to scaling the original Celsius values. Specifically, we apply the factor \(\frac{9}{5}\) to the original standard deviation.
  • Given the standard deviation in Celsius is \(2^{\circ}C\), we utilize the equation \( SD_{F} = \left(\frac{9}{5}\right) \times SD_{C} \)
  • For our exercise, this becomes \( SD_{F} = \left(\frac{9}{5}\right) \times 2 = 3.6\).
Thus, the standard deviation in Fahrenheit is \(3.6^{\circ}F\), reflecting how spreading values scale under a linear transformation without being shifted by adding constants.
Celsius to Fahrenheit Conversion
Converting temperatures from Celsius to Fahrenheit is a common task in statistical analysis, particularly when working with data across different regions or reports using different scales. The formula
  • \(^\circ F = \left(\frac{9}{5}\right) \times ^\circ C + 32\)
is derived from the linear relationship between the two scales. For an individual value, it's essential to remember:
  • The multiplication by \(\frac{9}{5}\) adjusts the unit size. Celsius has 100 units between freezing and boiling, while Fahrenheit has 180.
  • With the addition of 32, the two scales align, fixing the zero-points' discrepancy, since freezing in Celsius is 0, whereas in Fahrenheit it is 32.
Performing such conversions is fundamental, whether it's for weather reports or global scientific research, ensuring consistent and comparable data presentation.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Length of pregnancies The length of human pregnancies from conception to birth varies according to a distribution that is approximately Normal with mean 266 days and standard deviation 16 days. (a) At what percentile is a pregnancy that lasts 240 days (that's about 8 months)? (b) What percent of pregnancies last between 240 and 270 days (roughly between 8 months and 9 months)? (c) How long do the longest \(20 \%\) of pregnancies last?

Deciles The deciles of any distribution are the values at the 10 th, \(20 t h, \ldots, 90\) th percentiles. The first and last deciles are the l0th and the 90 th percentiles, respectively. (a) What are the first and last deciles of the standard Normal distribution? (b) The heights of young women are approximately Normal with mean 64.5 inches and standard deviation 2.5 inches. What are the first and last deciles of this distribution? Show your work.

Density of the earth In 1798 , the English scientist Henry Cavendish measured the density of the earth several times by careful work with a torsion balance. The variable recorded was the density of the earth as a multiple of the density of water. Here are Cavendish's 29 measurements: \({ }^{12}\) $$ \begin{array}{llllllllll} \hline 5.50 & 5.61 & 4.88 & 5.07 & 5.26 & 5.55 & 5.36 & 5.29 & 5.58 & 5.65 \\ 5.57 & 5.53 & 5.62 & 5.29 & 5.44 & 5.34 & 5.79 & 5.10 & 5.27 & 5.39 \\ 5.42 & 5.47 & 5.63 & 5.34 & 5.46 & 5.30 & 5.75 & 5.68 & 5.85 & \\ \hline \end{array} $$ (a) Enter these data into your calculator and make a histogram. Include a sketch of the graph on your paper. Then calculate one-variable statistics. Describe the shape, center, and spread of the distribution of density measurements. (b) Calculate the percent of observations that fall within \(1,2,\) and 3 standard deviations of the mean. How do these results compare with the \(68-95-99.7\) rule? (c) Use your calculator to construct a Normal probability plot. Include a sketch of the graph on your paper. Interpret this plot. (d) Having inspected the data from several different perspectives, do you think these data are approximately Normal? Write a brief summary of your assessment that combines your findings from parts (a) through (c).

Men's heights The distribution of heights of adult American men is approximately Normal with mean 69 inches and standard deviation 2.5 inches. Draw an accurate sketch of the distribution of men's heights. Be sure to label the mean, as well as the points 1,2 , and 3 standard deviations away from the mean on the horizontal axis.

Scores on the \(A C\) T college entrance exam follow a bell-shaped distribution with mean 18 and standard deviation 6 . Wayne's standardized score on the ACT was \(-0.5 .\) What was Wayne's actual ACT score? (a) 5.5 (b) 12 (c) 15 (d) 17.5 (e) 21
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Mechanics Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.