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Think and Answer requires you to answer two questions

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Think and Answer requires you to answer two questions. For each question, do not simply provide an answer; make sure you explain how you arrived at that answer. Even if your reasoning is wrong, you will still be credited for participation.

Answer the following question:

1. We define probabilities for various normal distributions. However, we use one single table (Standard Normal Distribution Table) for all of them. Explain why that is possible.

True or False?

2. The more lottery tickets you buy, the higher is your chance to win a prize.

3. Health insurance policy is designed in such a way that a healthy person has a negative expected value, and an ailing one has a positive expected value.

4. Because the tails of the normal distribution curve are infinitely long, the total area under the curve is also infinite.

HW 3

Ch. 5

Sec 5-1:

1. Random Variable The accompanying table lists probabilities for the corresponding numbers of girls in four births. What is the random variable, what are its possible values, and are its values numerical?
   Number of Girls in Four Births
    

Number of Girls x	P(x)
0	0.063
1	0.250
2	0.375
3	0.250
4	0.063

2.  

 

3. Discrete or Continuous? Is the random variable given in the accompanying table discrete or continuous? Explain.

 

Identifying Probability Distributions In Exercises 7–14, determine whether a probability distribution is given. If a probability distribution is given, find its mean and standard deviation. If a probability distribution is not given, identify the requirements that are not satisfied.

9. Pickup Line Ted is not particularly creative. He uses the pickup line “If I could rearrange the alphabet, I’d put U and I together.” The random variable x is the number of women Ted approaches before encountering one who reacts positively.

x	P(x)
1	0.001
2	0.009
3	0.030
4	0.060

 

 

10. Fun Ways to Flirt In a Microsoft Instant Messaging survey, respondents were asked to choose the most fun way to flirt, and the accompanying table is based on the results.

	P(x)
E-mail	0.06
In person	0.55
Instant message	0.24
Text message	0.15

 

 

 

13 (a,c)

Cell Phone Use In a survey, cell phone users were asked which ear they use to hear their cell phone, and the table is based on their responses (based on data from “Hemispheric Dominance and Cell Phone Use,” by Seidman et al., JAMA Otolaryngology—Head & Neck Surgery, Vol. 139, No. 5).

	P(x)
Left	0.636
Right	0.304
No preference	0.060

 

 

Genetics. In Exercises 15–20, refer to the accompanying table, which describes results from groups of 8 births from 8 different sets of parents. The random variable x represents the number of girls among 8 children.

Number of Girls x	P(x)
0	0.004
1	0.031
2	0.109
3	0.219
4	0.273
5	0.219
6	0.109
7	0.031
8	0.004

 

 

 

15. Mean and Standard Deviation Find the mean and standard deviation for the numbers of girls in 8 births.
     

16. Range Rule of Thumb for Significant Events Use the range rule of thumb to determine whether 1 girl in 8 births is a significantly low number of girls.
 

18.Using Probabilities for Significant Events

• Find the probability of getting exactly 7 girls in 8 births.
• Find the probability of getting 7 or more girls in 8 births.
• Which probability is relevant for determining whether 7 is a significantly high number of girls in 10 births: the result from part (a) or part (b)?
• Is 7 a significantly high number of girls in 8 births? Why or why not?

 

 

 

 

28. Expected Value in Roulette When playing roulette at the Venetian casino in Las Vegas, a gambler is trying to decide whether to bet $5 on the number 27 or to bet $5 that the outcome is any one of these five possibilities: 0, 00, 1, 2, 3. From Example 6, we know that the expected value of the $5 bet for a single number is −26
c (slash through it). For the $5 bet that the outcome is 0, 00, 1, 2, or 3, there is a probability of 5/38 of making a net profit of $30 and a 33/38 probability of losing $5.

a. Find the expected value for the $5 bet that the outcome is 0, 00, 1, 2, or 3.

b. Which bet is better: a $5 bet on the number 27 or a $5 bet that the outcome is any one of the numbers 0, 00, 1, 2, or 3? Why?

 

29. Expected Value for Life Insurance There is a 0.9986 probability that a randomly selected 30-year-old male lives through the year (based on data from the U.S. Department of Health and Human Services). A Fidelity life insurance company charges $161 for insuring that the male will live through the year. If the male does not survive the year, the policy pays out $100,000 as a death benefit.

a.  From the perspective of the 30-year-old male, what are the monetary values corresponding to the two events of surviving the year and not surviving?

b. If a 30-year-old male purchases the policy, what is his expected value?

c. Can the insurance company expect to make a profit from many such policies? Why?

Ch. 6

Sec.6-1:

1. Normal Distribution What’s wrong with the following statement? “Because the digits 0, 1, 2, . . ., 9 are the normal results from lottery drawings, such randomly selected numbers have a normal distribution.”
    
2. Normal Distribution A normal distribution is informally described as a probability distribution that is “bell-shaped” when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.

 

Standard Normal Distribution. In Exercises 9–12, find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

 

9. 

    
   
    
10.  

    

     
    
     
11.  

    

     
    
     
12.  

    

 

Standard Normal Distribution. In Exercises 13–16, find the indicated z score. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.

 

13. 

     
    
     
14.  

    

 

Standard Normal Distribution. In Exercises 17–36, assume that a randomly selected subject is given a bone density test. Those test scores are normally distributed with a mean of 0 and a standard deviation of 1. In each case, draw a graph, then find the probability of the given bone density test scores. If using technology instead of Table A-2, round answers to four decimal places.

 

18. Less than −1.96

 25. Between 2.00 and 3.00

28. Between −2.75 and −0.75

 

Critical Values. In Exercises 41–44, find the indicated critical value. Round results to two decimal places.

 

41, z0.10

 

Sec. 6-3:

 

1. Births There are about 11,000 births each day in the United States, and the proportion of boys born in the United States is 0.512. Assume that each day, 100 births are randomly selected and the proportion of boys is recorded.
   1. What do you know about the mean of the sample proportions?
   2. What do you know about the shape of the distribution of the sample proportions?

 

2. Sampling with Replacement The Orangetown Medical Research Center randomly selects 100 births in the United States each day, and the proportion of boys is recorded for each sample.
   3. Do you think the births are randomly selected with replacement or without replacement?
   4. Give two reasons why statistical methods tend to be based on the assumption that sampling is conducted with replacement, instead of without replacement.

 

 

4. Sampling Distribution Data Set 4 “Births” in Appendix B includes a sample of birth weights. If we explore this sample of 400 birth weights by constructing a histogram and finding the mean and standard deviation, do those results describe the sampling distribution of the mean? Why or why not?

 

6. College Presidents There are about 4200 college presidents in the United States, and they have annual incomes with a distribution that is skewed instead of being normal. Many different samples of 40 college presidents are randomly selected, and the mean annual income is computed for each sample.

1. What is the approximate shape of the distribution of the sample means (uniform, normal, skewed, other)?
2. What value do the sample means target? That is, what is the mean of all such sample means?

 

 

Sec 6-4:

 

Interpreting Normal Quantile Plots. In Exercises 5–8, examine the normal quantile plot and determine whether the sample data appear to be from a population with a normal distribution.

5. Ages of Presidents The normal quantile plot represents the ages of presidents of the United States at the times of their inaugurations. The data are from Data Set 15 “Presidents” in Appendix B.

 

 

 

 

6. Diet Pepsi The normal quantile plot represents weights (pounds) of the contents of cans of Diet Pepsi from Data Set 26 “Cola Weights and Volumes” in Appendix B.

 

Determining Normality. In Exercises 9–12, refer to the indicated sample data and determine whether they appear to be from a population with a normal distribution. Assume that this requirement is loose in the sense that the population distribution need not be exactly normal, but it must be a distribution that is roughly bell-shaped.

 

 

9.

cookies The numbers of chocolate chips in Chips Ahoy (reduced fat) cookies, as listed in Data Set 28 “Chocolate Chip Cookies” in Appendix B.

 

 

 

Using Technology to Generate Normal Quantile Plots. In Exercises 13–16, use the data from the indicated exercise in this section. Use software (such as Statdisk, Minitab, Excel, or StatCrunch) or a TI-83/84 Plus calculator to generate a normal quantile plot. Then determine whether the data come from a normally distributed population.

 

 14,

Exercise 11 “Garbage”