CHAPTER 4 TEST REVIEW

Probability & Statistics

Decide whether the random variable is discrete or continuous.

X represents the number of motorcycle accidents in one year in California.

X represents the volume of blood drawn for a blood test.

Discrete

Continuous

A company gave psychological tests to prospective employees. The random variable represents the possible test scores. Use the histogram to find the probability that a person selected at random from the survey’s sample had a test score of (a) more than two and (b) less than four.

𝑷 (𝒕𝒆𝒔𝒕 𝒔𝒄𝒐𝒓𝒆>𝟐 )=𝟎 .𝟑𝟓

𝑷 (𝒕𝒆𝒔𝒕 𝒔𝒄𝒐𝒓𝒆<𝟒 )=𝟎 .𝟗𝟎

Determine the probability distribution’s missing probability value.

A sociologist surveyed the households in a small town. The random variable represents the number of dependent children in the households.

𝑷 (𝟑𝒅𝒆𝒑𝒆𝒏𝒅𝒆𝒏𝒕𝒄𝒉𝒊𝒍𝒅𝒓𝒆𝒏 )=𝟎 .𝟐𝟐

Use the frequency distribution to construct a probability distribution, find the mean, variance, and standard deviation of the probability distribution. Round all numbers to the nearest third decimal place except for the standard deviation (nearest tenth).

The following is a probability distribution of the number of dogs found in the respective households in a small town. Complete the table and missing values.

households

0 1466

1 582

2 329

3 165

4 96Frequency =

Mean =

Variance =

Standard deviation =∑ 𝑃 (𝑥 )=¿¿Expected value =

Use the frequency distribution to construct a probability distribution, find the mean, variance, and standard deviation of the probability distribution. Round all numbers to the nearest third decimal place except for the standard deviation (nearest tenth).The following is a probability distribution of the number of dogs found in the respective households in a small town. Complete the table and missing values.household

s

0 1466 0.556 0.000 0.646 0.359

1 582 0.221 0.221 0.196 0.038 0.008

2 329 0.125 0.250 1.196 1.430 0.179

3 165 0.063 0.189 2.196 4.822 0.304

4 96 0.036 0.144 3.196 10.214 0.368

Frequency = 2638

Mean = 0.804

Variance = 1.218

Standard deviation = 1.1∑ 𝑃 (𝑥 )=𝟏 .𝟎𝟎𝟏≈𝟏Expected value = 0.804

A binomial experiment is a probability experiment that satisfies the following conditions.

B Binary There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success or as a failure.

I Independent Each trial is independent of the other trials.

N Number The experiment is repeated for a fixed number of trials

S Success The probability of a success is the same for each trial.

You are taking a multiple-choice quiz that consists of five questions. Each question has four possible answers, only one of which is correct. To complete the quiz, you randomly guess the answer to each question. Find the probability of guessing (a) exactly three answers correctly,(b) at least three answers correctly, and (c) less than three answers correctly. Use the formula to get your answers.

nCx

5C0

𝑛=5𝑝=0.25𝑞=0.75𝑥=0 ,1 ,2 ,3 , 4 ,5

5C1

5C2

5C3

5C0

5C1

5C2

5C3

𝑷 (𝒆𝒙𝒂𝒄𝒕𝒍𝒚 𝒕𝒉𝒓𝒆𝒆𝒂𝒏𝒔𝒘𝒆𝒓𝒔𝒄𝒐𝒓𝒓𝒆𝒄𝒕𝒍𝒚 )=𝟎 .𝟎𝟖𝟖

less than three answers correctly)¿𝟎 .𝟐𝟑𝟕+𝟎 .𝟑𝟗𝟔+𝟎 .𝟐𝟔𝟒=𝟎 .𝟖𝟗𝟕

𝑷 (𝒂𝒕 𝒍𝒆𝒂𝒔𝒕 𝒕𝒉𝒓𝒆𝒆𝒂𝒏𝒔𝒘𝒆𝒓𝒔𝒄𝒐𝒓𝒓𝒆𝒄𝒕𝒍𝒚 )=𝟏−𝟎 .𝟖𝟗𝟕=𝟎 .𝟏𝟎𝟑

Fourteen percent of workers believe they will need less than $250,000 when they retire. You randomly select 10 workers and ask each how much money he or she thinks they will need for retirement. Find the probability that the number of workers who say they will need less than $250,000 when they retire is (a) exactly two, and (b) at most five. Use the graphing calculator to get your answers.

𝑃 (𝑥=0 )=0.221

𝑃 (𝑥=1 )=0.360

𝑃 (𝑥=2 )=0.264

𝑃 (𝑥=3 )=0.115

𝑃 (𝑥=4 )=0.033

𝑃 (𝑥=5 )=0.006

𝑷 (𝒆𝒙𝒂𝒄𝒕𝒍𝒚 𝒕𝒘𝒐𝒘𝒐𝒓𝒌𝒆𝒓𝒔 )=𝟎 .𝟐𝟔𝟒

𝑷 (𝒂𝒕𝒎𝒐𝒔𝒕𝟓𝒘𝒐𝒓𝒌𝒆𝒓𝒔 )=𝑷 (𝟎 )+𝑷 (𝟏 )+𝑷 (𝟐 )+𝑷 (𝟑 )+𝑷 (𝟒 )+𝑷 (𝟓 )

¿ (𝟎 .𝟐𝟐𝟏+𝟎 .𝟑𝟔𝟎+𝟎 .𝟐𝟔𝟒+𝟎 .𝟏𝟏𝟓+𝟎 .𝟎𝟑𝟑+𝟎 .𝟎𝟎𝟔)

¿𝟎 .𝟗𝟗𝟗

1. Construct a binomial distribution2. Graph the binomial distribution using a histogram3. Find the mean, variance, and standard deviation of the binomial

distribution.

Thirty-seven percent of women consider themselves fans of professional baseball. You randomly select six women and ask each if she considers herself a fan of professional baseball.

𝑛=6𝑝=0.37𝑞=0.63 𝑥=0 ,1 ,2 ,3 ,4 ,5 ,6

Use technology to help construct a binomial distribution (table).

Go to 2nd Distr.Scroll down to binompdfTrials = 6

Paste to the home screen, then Enter.

0

1

2

3

4

5

6

Construct a binomial distribution

Go to 2nd Distr.Scroll down to binompdfTrials = 6

Paste to the home screen, then Enter

0 0.063

1 0.220

2 0.323

3 0.253

4 0.112

5 0.026

6 0.003

Are all the probabilities between 0 and 1?

yes

Add up the probabilities.

∑ 𝒑 (𝒙 )=𝟏

The random variable x is on the x-axis. Place the numbers below the middle of each bar.

Label both x-axis and y-axis.

The y-axis is probability.

Title the graph

Use increments of 0.05 on the y-axis.

Make sure the bars touch.

The bars must be of equal width.

Graph the binomial distribution using a histogram

Find the mean, variance, and standard deviation of the binomial distribution. Round the mean and variance to three decimal places, and round the standard deviation to the nearest tenth.

𝑚𝑒𝑎𝑛=𝜇=𝑛𝑝=(6 ) (0.37 )=𝟐 .𝟐𝟐

We need to know the values for

𝑛=6𝑝=0.37𝑞=0.63

𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒=𝜎 2=𝑛𝑝𝑞= (6 ) (0.37 ) (0.63 )≈𝟏 .𝟑𝟗𝟗

𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛=𝜎=√𝑛𝑝𝑞≈𝟏 .𝟐

Basketball player Shaquille O’Neal makes a free-throw shot about 52.6% of the time. Find the probability that (a) the first shot O’Neal makes is the second shot, (b) the first shot O’Neal makes is the first or second shot.

This is a geometric probability and we use the formula

249

𝑝=0.526𝑞=0.474

𝑝 ( 𝑓𝑖𝑟𝑠𝑡 𝑜𝑟 𝑠𝑒𝑐𝑜𝑛𝑑 h𝑠 𝑜𝑡 )=𝑝 ( 𝑓𝑖𝑟𝑠𝑡 h𝑠 𝑜𝑡 )+𝑝 (𝑠𝑒𝑐𝑜𝑛𝑑 h𝑠 𝑜𝑡)

𝑝 ( 𝑓𝑖𝑟𝑠𝑡 h𝑠 𝑜𝑡 )=(0.526 ) (0.474 )1−1= (0.526 ) (0.474 )0= (0.526 ) (1 )=0.526

𝑝 (𝑠𝑒𝑐𝑜𝑛𝑑 h𝑠 𝑜𝑡 )≈0.249

𝒑 ( 𝒇𝒊𝒓𝒔𝒕𝒐𝒓 𝒔𝒆𝒄𝒐𝒏𝒅𝒔𝒉𝒐𝒕 )=𝟎 .𝟓𝟐𝟔+𝟎 .𝟐𝟒𝟗=𝟎 .𝟕𝟕𝟓