Understanding the Normal Distribution: A Key Concept in Science, Finance, and Economics | Numerade (2024)

The average amount of precipitation in Dallas, Texas, during the month of April is 3.5 inches (The World Almanac, 2000 ). Assume that a normal distribution applies and that the standard deviation is .8 inches.
a. What percentage of the time does the amount of rainfall in April exceed 5 inches?
b. What percentage of the time is the amount of rainfall in April less than 3 inches?
c. $A$ month is classified as extremely wet if the amount of rainfall is in the upper $10 \%$ for that month. How much precipitation must fall in April for it to be classified as extremely wet?

Continuous Probability Distributions

Draw a graph for the standard normal distribution. Label the horizontal axis at values of $-3,-2,-1,0,1,2,$ and $3 .$ Then use the table of probabilities for the standard normal distribution inside the front cover of the text to compute the following probabilities.
a. $ P(z \leq 1.5)$
b. $ P(z \leq 1)$
c. $P(1 \leq z \leq 1.5)$
d. $P(0 < z < 2.5)$

Continuous Probability Distributions

Draw a graph for the standard normal distribution. Label the horizontal axis at values of $-3,-2,-1,0,1,2,$ and $3 .$ Then use the table of probabilities for the standard normal distribution inside the front cover of the text to compute the following probabilities.
a. $ P(z \leq 1.5)$
b. $ P(z \leq 1)$
c. $P(1 \leq z \leq 1.5)$
d. $P(0 < z < 2.5)$

Continuous Probability Distributions

Refer to Exercise $7.11 .$ Suppose that in the forest fertilization problem the population standard deviation of basal areas is not known and must be estimated from the sample. If a random sample of $n=9$ basal areas is to be measured, find two statistics $g_{1}$ and $g_{2}$ such that $P\left[g_{1} \leq(\bar{Y}-\mu) \leq g_{2}\right]=.90$.

Sampling Distributions and the Central Limit Theorem

Sampling Distributions Related to the Normal Distribution

a. Use the applet Chi-Square Probabilities and Quantiles to find $P[Y>E(Y)]$ when $Y$ has $\chi^{2}$ distributions with $10,40,$ and 80 df.
b. What did you notice about $P[Y>E(Y)]$ as the number of degrees of freedom increases as in part (a)?
c. How does what you observed in part (b) relate to the shapes of the $\chi^{2}$ densities that you obtained in Exercise $7.22 ?$

Sampling Distributions and the Central Limit Theorem

Sampling Distributions Related to the Normal Distribution

Ammeters produced by a manufacturer are marketed under the specification that the standard deviation of gauge readings is no larger than .2 amp. One of these ammeters was used to make ten independent readings on a test circuit with constant current. If the sample variance of these ten measurements is. 065 and it is reasonable to assume that the readings are normally distributed, do the results suggest that the ammeter used does not meet the marketing specifications? [Hint: Find the approximate probability that the sample variance will exceed. 065 if the true population variance is .04.]

Sampling Distributions and the Central Limit Theorem

Sampling Distributions Related to the Normal Distribution

A _________ ________ _________is an equation used to compute probabilities of continuous random variables.

The Normal Probability Distribution

Properties of the Normal Distribution

There is no nice formula for the standard normal cdf $\Phi(z),$ but several good approximations have been published in articles. The following is from "Approximations for Hand Calculators Using Small Integer Coefficients" (Math. Comput, $1977 : 214-222 ) .$ For $0 < z \leq 5.5$ ,
$$P(Z \geq z)=1-\Phi(z) \approx .5 \exp \left\{-\left[\frac{(83 z+351) z+562}{(703 / z)+165}\right]\right\}$$
The relative error of this approximation is less than .042$\% .$ Use this to calculate approximations to the following probabilities, and compare whenever possible to the probabilities obtained from Appendix Table A.3.
(a) $P(Z \geq 1)$
(b) $P(Z<-3)$
(c) $P(-4< Z<4)$
(d) $P(Z > 5)$

Continuous Random Variables and Probability Distributions

The Normal (Gaussian) Distribution

There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 $\mathrm{cm}$ and standard deviation .1 $\mathrm{cm} .$ The second machine produces corks with diameters that have a normal distribution with mean 3.04 $\mathrm{cm}$ and standard deviation .02 $\mathrm{cm} .$ Acceptable corks have diameters between 2.9 and 3.1 $\mathrm{cm} .$ Which machine is more likely to produce an acceptable cork?

Continuous Random Variables and Probability Distributions

The Normal (Gaussian) Distribution

Use a normal probability plot to assess whether the sample data could have come from a population that is normally distributed. A random sample of 20 undergraduate students receiving student loans was obtained, and the amount of their loans for the $2011-2012$ school year was recorded.
$$\begin{array}{lllll}2,500 & 1,000 & 2,000 & 14,000 & 1,800 \\\hline 3,800 & 10,100 & 2,200 & 900 & 1,600 \\
\hline 500 & 2,200 & 6,200 & 9,100 & 2,800 \\\hline 2,500 & 1,400 & 13,200 & 750 & 12,000\end{array}$$

The Normal Probability Distribution

Assessing Normality

The time required for Speedy Lube to complete an oil change service on an automobile approximately follows a normal distribution, with a mean of 17 minutes and a standard deviation of 2.5 minutes.
(a) Speedy Lube guarantees customers that the service will take no longer than 20 minutes. If it does take longer, the customer will receive the service for half-price. What percent of customers receive the service for half price?
(b) If Speedy Lube does not want to give the discount to more than $3 \%$ of its customers, how long should it make the guaranteed time limit?

The Normal Probability Distribution

Applications of the Normal Distribution

See Problem 45. In games where a team is favored by more than 12 points, the margin of victory for the favored team relative to the spread is normally distributed with a mean of -1.0 point and a standard deviation of 10.9 points. Source: Justin Wolfers, "Point Shaving: Corruption in NCAA Basketball"
(a) In games where a team is favored by more than 12 points, what is the probability that the favored team wins by 5 or more points relative to the spread?
(b) In games where a team is favored by more than 12 points, what is the probability that the favored team loses by 2 or more points relative to the spread?
(c) In games where a team is favored by more than 12 points, what is the probability that the favored team "beats the spread"? Does this imply that the possible point shaving spreads are accurate for games in which a team is favored by more than
12 points?

The Normal Probability Distribution

Applications of the Normal Distribution

What is the mean square due to treatment estimate of $\sigma^{2} ?$ What is the mean square due to error estimate of $\sigma^{2} ?$

Comparing Three or More Means

Comparing Three or More Means

The acronym $ANOVA$ stands for _____ _ _____.

Comparing Three or More Means

Comparing Three or More Means

One term at the University of California, Berkeley, 400 students took the final in Statistics 2 . Their scores averaged 65.3 out of $100,$ and the $S D$ was $25 .$ Now
$\sqrt{400} \times 25=500, \quad 500 / 400=1.25$
Is 65.3$\pm 2.5 a 95 \%$ -confidence interval? If so, for what? If not, why not?

The Accuracy of Averages

Cycle Time A cement truck delivers mixed cement to a large construction site. Let $x$ represent the cycle time in minutes for the truck to leave the construction site, go back to the cement plant, fill
up, and return to the construction site with another load of cement. From past experience, it is known that the mean cycle time is $\mu=45$ minutes with
$\sigma=12$ minutes. The $x$ distribution is approximately normal.
(a) What is the probability that the cycle time will exceed 60 minutes, given that it has exceeded 50 minutes? Hint: See Problem $39,$ part (c).
(b) What is the probability that the cycle time will exceed 55 minutes, given that it has exceeded 40 minutes?

Normal Curves and Sampling Distributions

Areas Under Any Normal Curve

Estimating the Standard Deviation Consumer Reports gave information about the ages at which various household products are replaced. For example, color TVs are replaced at an average age of $\mu=8$ years after purchase, and the (95\% of data) range was from 5 to 11 years. Thus, the range was $11-5=6$ years. Let $x$ be the age (in years) at which a color
TV is replaced. Assume that $x$ has a distribution that is approximately normal.
(a) The empirical rule (see Section 6.1) indicates that for a symmetric and bell-shaped distribution, approximately $95 \%$ of the data lies within two standard deviations of the mean. Therefore, a $95 \%$ range of data values extending from $\mu-2 \sigma$ to $\mu+2 \sigma$ is often used for "commonly occurring" data values. Note that the interval from $\mu-2 \sigma$ to $\mu+2 \sigma$ is $4 \sigma$ in length. This leads to a "rule of thumb" for estimating the standard deviation from a $95 \%$ range of data values.Use this "rule of thumb" to approximate the standard deviation of $x$ values, where $x$ is the age (in years) at which a color TV is replaced.
(b) What is the probability that someone will keep a color TV more than
5 years before replacement?
(c) What is the probability that someone will keep a color TV fewer than
10 years before replacement?
(d) Inverse Normal Distribution Assume that the average life of a color
TV is 8 years with a standard deviation of 1.5 years before it breaks. Suppose that a company guarantees color TVs and will replace a
TV that breaks while under guarantee with a new one. However, the company does not want to replace more than $10 \%$ of the TVs under guarantee. For how long should the guarantee be made (rounded to the nearest tenth of a year)?

Normal Curves and Sampling Distributions

Areas Under Any Normal Curve

Law Enforcement: Police Response Time Police response time to an emergency call is the difference between the time the call is first received by the dispatcher and the time a patrol car radios that it has arrived at the scene (based on information from The Denver Post). Over a long period of time, it has been determined that the police response time has a normal distribution with a mean of 8.4 minutes and a standard deviation of 1.7 minutes. For a randomly received emergency call, what is the probability that the response time will be
(a) between 5 and 10 minutes?
(b) less than 5 minutes?
(c) more than 10 minutes?

Normal Curves and Sampling Distributions

Areas Under Any Normal Curve

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.
Aircraft Altimeters The Skytek Avionics company uses a new production method to manufacture aircraft altimeters. A simple random sample of new altimeters resulted in the errors listed below. Use a 0.05 level of significance to test the claim that the new production method has errors with a standard deviation greater than $32.2 \mathrm{ft}$, which was the standard deviation for the old production method. If it appears that the standard deviation is greater, does the new production method appear to be better or worse than the old method? Should the company take any action? $$\begin{array}{ccccccccccc} -42 & 78 & -22 & -72 & -45 & 15 & 17 & 51 & -5 & -53 & -9 & -109 \end{array}$$

Hypothesis Testing

Testing a Claim About a Standard Deviation or Variance

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, or critical value(s), then state the conclusion about the null hypothesis, as well as the final conclusion that addresses the original claim. Assume that a simple random sample is selected from a normally distributed population.
Pulse Rates of Men A simple random sample of 153 men results in a standard deviation of 11.3 beats per minute (based on Data Set 1 "Body Data" in Appendix B). The normal range of pulse rates of adults is typically given as 60 to 100 beats per minute. If the range rule of thumb is applied to that normal range, the result is a standard deviation of 10 beats per minute. Use the sample results with a 0.05 significance level to test the claim that pulse rates of men have a standard deviation equal to 10 beats per minute; see the accompanying StatCrunch display for this test. What do the results indicate about the effectiveness of using the range rule of thumb with the "normal range" from 60 to 100 beats per minute for estimating $\sigma$ in this case?
(GRAPH CAN'T COPY)

Hypothesis Testing

Testing a Claim About a Standard Deviation or Variance

SAT and ACT Tests Because they enable efficient procedures for evaluating answers, multiple choice questions are commonly used on standardized tests, such as the SAT or ACT. Such questions typically have five choices, one of which is correct. Assume that you must make random guesses for two such questions. Assume that both questions have correct answers of "a."
a. After listing the 25 different possible samples, find the proportion of correct answers in each sample, then construct a table that describes the sampling distribution of the sample proportions of correct responses.
b. Find the mean of the sampling distribution of the sample proportion.
c. Is the mean of the sampling distribution [from part (b)] equal to the population proportion of correct responses? Does the mean of the sampling distribution of proportions always equal the population proportion?

Normal Probability Distributions

Sampling Distributions and Estimators

Use the following information. A drug manufacturer claims that a drug cures a rare skin disease $75 \%$ of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted.
Find the probability that the claim will be accepted assuming that the actual probability that the drug cures the skin disease is $65 \% .$

Normal Probability Distributions

Normal Approximations to Binomial Distributions

Use the following information. A drug manufacturer claims that a drug cures a rare skin disease $75 \%$ of the time. The claim is checked by testing the drug on 100 patients. If at least 70 patients are cured, then this claim will be accepted.
Find the probability that the claim will be rejected assuming that the manufacturer's claim is true.

Normal Probability Distributions

Normal Approximations to Binomial Distributions

Five percent of U.S. workers use public transportation to get to work. A transit authority offers discount rates to companies that have at least 30 employees who use public transportation to get to work. Find the probability that each company will get the discount. (Source U.S. Census Bureau)
(a) Company A has 250 employees.
(b) Company B has 500 employees.
(c) Company $\mathrm{C}$ has 1000 employees.

Normal Probability Distributions

Normal Approximations to Binomial Distributions

Finding Critical Values of $x^{2}$ Repeat Exercise 19 using this approximation (with $k$ and $z$ as described in Exercise 19): $$\chi^{2}=k\left(1-\frac{2}{9 k}+z \sqrt{\frac{2}{9 k}}\right)^{3}$$

Hypothesis Testing

Testing a Claim About a Standard Deviation or Variance

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
Researchers collected data on the numbers of hospital admissions resulting from motor vehicle crashes, and results are given below for Fridays on the 6 th of a month and Fridays on the following 13 th of the same month (based on data from "Is Friday the 13 th Bad for Your Health?" by Scanlon et al., British Medical Journal, Vol. $307,$ as listed in the Data and Story Line online resource of data sets). Construct a $95 \%$ confidence interval estimate of the mean of the population of differences between hospital admissions on days that are Friday the 6 th of a month and days that are Friday the 13 th of a month. Use the confidence interval
to test the claim that when the 13 th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected.
$$\begin{array}{|l|c|c|c|c|c|c|}
\hline \text { Friday the 6th } & 9 & 6 & 11 & 11 & 3 & 5 \\
\hline \text { Friday the 13th } & 13 & 12 & 14 & 10 & 4 & 12 \\
\hline
\end{array}$$

Inferences from Two Samples

Two Dependent Samples

Use the listed paired sample data, and assume that the samples are simple random samples and that the differences have a distribution that is approximately normal.
A popular theory is that presidential candidates have an advantage if they are taller than their main opponents. Listed are heights (cm) of presidents along with the heights of their main opponents (from Data Set 15 "Presidents").
a. Use the sample data with a 0.05 significance level to test the claim that for the population of heights of presidents and their main opponents, the differences have a mean greater than $0 \mathrm{cm} .$
b. Construct the confidence interval that could be used for the hypothesis test described in part
(a). What feature of the confidence interval leads to the same conclusion reached in part (a)?
$$\begin{array}{|l|l|l|l|l|l|l|}
\hline \text { Height (cm) of President } & 185 & 178 & 175 & 183 & 193 & 173 \\
\hline \text { Height (cm) of Main Opponent } & 171 & 180 & 173 & 175 & 188 & 178 \\
\hline
\end{array}$$

Inferences from Two Samples

Two Dependent Samples