Help:Lesson 8 Print

From BYUI Statistics Text

The following questions are intended to help you judge your preparation for this exam. Carefully work through theproblems.These questions are repeated on the preparation quiz for this lesson.

This is not designed to be a comprehensive review. There may be items on the exam that are not covered in thisreview. Similarly, there may be items in this review that are not tested on this exam. You are strongly encouraged toreview the readings, homework exercises, and other activities from Unit 1 as you prepare for the exam. Use theINDEX to review definitions of important terms.

1 Lesson Summaries

Lesson 01 Recap

Each lesson follows the same schedule: Individual and Group Preparation, Class

Meeting, and Homework Assignment and Quiz. Understanding this layout willhelp you successfully manage the workload of this class.

In this class you will use the online textbook that has been written for you by yourstatistics teachers. All of the assignments and quizzes will be based on the readings,so study it well.

By doing the work, staying on schedule, and living the Honor Code you can succeedin this class!

Lesson 02 Recap

The Statistical Process has five steps: Design the study, Collect the data, Describe

the data, Make inferences, Take action.

In a designed experiment, researchers control the conditions of the study. In an

observational study, researchers don't control the conditions but only observe whathappens.

There are many sampling methods used to obtain a sample from a population. The

most important is a simple random sample (SRS) which is a random selection takenfrom a population.

Quantitative variables represent things that are numeric in nature, such as the value

of a car or the number of students in a classroom. Categorical variables represent

nonnumerical data that can only be considered as labels, such as colors or brands ofshoes.

The null hypothesis ( ) is the foundational assumption about a population and

represents the status quo. The alternative hypothesis ( ) is a different

assumption about a population. Using a hypothesis test, we determine whether it ismore likely that the null hypothesis or the alternative hypothesis is true.

Lesson 03 Recap

A histogram allows us to visually interpret data. Histograms can be left-skewed,right-skewed, or symmetrical and bell-shaped.

The mean, median, and mode are measures of the center of a distribution. Themean is the most common measure of center, and is computed by adding up theobserved data and dividing by the number of observations in the data set.

The standard deviation is a number that describes how spread out the data are. Alarger standard deviation means the data are more spread out than data with a smallerstandard deviation.

A parameter is a true (but usually unknown) number that describes a population. A

statistic is an estimate of a parameter obtained from a sample.

Quartiles/percentiles , Five-Number Summaries , and Boxplots are tools thathelp us understand data. The five-number summary of a data set contains theminimum value, the first quartile, the median, the third quartile, and the maximumvalue. A boxplot is a graphical representation of the five-number summary.

Lesson 04 Recap

The three rules of probability are:

1. A probability is a number between 0 and 1.

2. If you list all the outcomes of a probability experiment (such as rolling a die) theprobability that one of these outcomes will occur is 1. In other words, the sum of theprobabilities in any probability is 1.

3. The probability that an outcome will not occur is 1 minus the probability that it will

Ho

Ha

0 ≤ P(X) ≤ 1

∑P(X) = 1

occur.

Lesson 05 Recap

A normal density curve is symmetric and bell-shaped. The curve lies above thehorizontal axis and the total area under the curve is equal to 1.

A standard normal distribution has a mean of 0 and a standard deviation of 1. The

68-95-99.7% rule states that when data are normally distributed, approximately68% of the data lie within 1 standard deviation from the mean, approximately 95% ofthe data lie within 2 standard deviations from the mean, and approximately 99.7% ofthe data lie within 3 standard deviations from the mean.

A z-score tells us how many standard deviations away from the mean a given value

is. It is calculated as:

The probability applet allows us to use z-scores to calculate proportions,probabilities, and percentiles.

A Q-Q plot is used to assess whether or not a set of data is normally distributed.

Lesson 06 Recap

The distribution of sample means is a distribution of all possible sample means ( )for a particular sample size. It has a mean of and a standard deviation of .

The distribution of sample means is normal when is normally distributed or when,

thanks to the Central Limit Theorom (CLT), our sample size ( ) is large.

The Law of Large Numbers states that as the sample size ( ) gets larger, thesample mean ( ) will get closer to the population mean ( ).

Lesson 07 Recap

When the distribution of sample means is normally distributed, we can use z-scores

and the probability applet to calculate proportions and probabilities. A z-score is

P(not X) = 1 − P(X)

z = =value − meanstandard deviation

x − μσ

x̄μ σ/ n−√

x̄n

nx̄ μ

z = =value − mean − μ¯

calculated as:

The -value is the probability of getting a test statistic at least as extreme as the oneyou got, assuming is true. A -value is calculated by finding the area under thenormal distribution curve that is more extreme (farther away from the mean) than thez-score.

2 Review Questions

1. What is the name of the important statistical result that guarantees that the sampling distribution of the samplemean will be normal, if the sample size is large?

2. What is the name of the important statistical result that states that when the sample size is large, the sample mean will be close to the population mean ?

3. Which of the following variables is/are categorical?

The number of animals in the local zoo

A randomly selected brand of toothpaste at a convenience store

Age in years of a randomly selected vehicle in a parking lot

4. The mean I.Q. test score in the United States is 100. Twenty randomly selected Statistics students took an I.Q.test, and the mean of their scores was 112. Which of these two numbers is a parameter and which is a statistic?

5. Rotham City has four distinct neighborhoods. The property values are very similar within each neighborhood, butthey vary considerably from neighborhood to neighborhood. One neighborhood has very low property values andanother has extremely high property values. Property taxes are usually roughly proportional to the property values.What type of sample should we collect to gauge the residents' responses to a proposed increase in property taxes?

6. When designed experiments are conducted in public school systems, researchers typically choose a particulardistrict, and randomly select teachers of, say, fourth grade students in that district. The experiment is thenconducted using all the students in the selected classes. Which type of sampling scheme is implemented?

7. At a school fund raiser, the name of every person who attended was entered into a drawing. At the end of theevening, six names were selected to receive door prizes. Which type of sampling scheme was implemented?

8. A special interest group is conducting a survey regarding wolves in Yellowstone National Park. To adequatelygauge opinions on both sides of the political spectrum, the researchers took the list of registered Democrats andselected a simple random sample of 100 people. Then, they took a list of registered Republicans and selected asimple random sample of 100 people. Which type of sampling scheme was implemented?

z = =value − meanstandard deviation

− μx̄σ/ n−√

PHo P

x̄

x̄ μ

I. II. III.

A survey was taken of purchases at the Crossroads. Open the data file CROSSROADSPURCHASES (Excel for 221 andSPSS for 222 and 223) and use this data to answer questions 9 and 10.

9. Calculate the descriptive statistics of the amounts spent in Crossroads purchases. Give the mean, standarddeviation, and five-number summary of the data.

10. Create a histogram representing the amounts spent in Crossroads purchases.

Jessica Meir and her research team measured the body temperatures of a sample of diving elephant seals. Athermistor was placed at a specific location on each seal to measure its body temperature. The body temperature ofseals tends to decrease as they dive. The researchers estimated the typical body temperature of each seal at thetime they initiate a dive and called this the "representative temperature" of the seal.

Thermistors were placed in the hepatic sinus of four of the seals (named Bodil, Roberta, Larry, and Per.) The seal'sbody temperature at the start of the dive was calculated to be:

SealRepresentative

Temperature ( )

Bodil 37.91

Roberta 37.25

Larry 38.98

Per 38.16

Complete the following table and answer questions 11 through 14 below.

Representative

Temperature ( )

Deviation from

the mean

Squared

Deviations

Seal

Bodil 37.91 "A"

Roberta 37.25 "B"

Larry 38.98

Per 38.16

11. What is the value of the number that goes in the position marked with an "A" in the table above?

12. What is the value of the number that goes in the position marked with an "B" in the table above?

13. What quantity is equal to the sum of the numbers in the "Squared Deviations" column?

C∘

C∘

x (x − )x̄ (x − x̄)2

A. The mean

B. The standard deviationC. The variance

D. None of these is correct

14. What is the standard deviation of these temperatures?

15. (True or False) Under certain conditions, the standard deviation can be negative.

16. In the same study, the researchers placed thermistors (temperature sensors) in the three seals (named Chick,Starburst, and Patty) near their spinal cord and brain in the extradural vein. They placed thermistors in six otherseals (named Sir Richard, Jerry, Sammy, Knut, Jonesie, and Butler) in their aorta, the largest artery in the body.Finally, they placed thermistors in four other seals (named Bodil, Roberta, Larry, and Per) in their hepatic sinus.The results from this study are summarized in the table below.

Sample

SizeMean

Standard

Deviation

Location

Arterial 6 38.785 1.554

Extradural 3 37.247 0.577

Hepatic Sinus 4 38.070 0.715

Which location resulted in the most consistent temperature measurements from one seal to another? Justify youranswer.

17. In your own words, and without using any mathematical symbols or statistical jargon, explain what the standarddeviation is. Do not tell how to calculate the standard deviation, but explain what it represents.

The number of hours students spent studying for an exam were recorded. The data are represented by the boxplotbelow. Use this boxplot to answer questions 18 and 19.

18. Give the five-number summary of this data.

n x̄ s

19. What percentage of the data lies between 4 hours and 8 hours?

An observation from a normally distributed population is considered "unusual" if it is more than 2 standarddeviations away from the mean. There are several contaminants that can harm a city's water supply. Nitrateconcentrations above 10 ppm (parts per million) are considered a health risk for infants less than six month of age.The City of Rexburg reports that the nitrate concentration in the city's drinking water supply is between 1.59 and2.52 ppm (parts per million,) and values outside of this range are unusual. We will assume 1.59 ppm is the value of

and is equal to 2.52 ppm. It is reasonable to assume the measured nitrate concentration is normally

distributed. Use this information to answer questions 20 through 22.

20. Estimate the mean of the measured nitrate concentration in Rexburg's drinking water.

21. Estimate the standard deviation of the measured nitrate concentration in Rexburg's drinking water.

22. Between what two measured nitrate concentrations do approximately 68% of the data values lie?

23. What is the mean and standard deviation of the standard normal distribution?

The admissions committees for most masters of business administration (MBA) programs require the GraduateManagement Admission Test (GMAT) as part of the application for new students. It has been shown that thescores on the GMAT are normally distributed with a mean of 542.3 and a standard deviation of 120.54. Theminimum GMAT score required for admission to the MAcc program in the School of Accountancy at BYU is 500,but it is rare for students with scores less than 600 to be admitted. Use this information to answer questions 24through 32.

24. What is the probability that a randomly selected student will score above 542.3 on the GMAT?

25. What is the probability that a randomly selected student will score below 600 on the GMAT?

26. What is the probability that a randomly selected student will score 600 or above on the GMAT?

27. Find the GMAT score that corresponds to the 10th percentile.

28. Find the GMAT score that corresponds to the 90th percentile.

29. Find the first quartile of the distribution of GMAT scores.

30. Find the third quartile of the distribution of GMAT scores.

μ − 2σ μ + 2σ

Consider a simple random sample of students taking the GMAT.

31. What are the mean and standard deviation of the distribution of sample mean scores for all such samples?

32. What is the probability that the sample mean GMAT score for a SRS of students will be greater than

600?

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n = 15

n = 15