Correct already written math assignment

In families with four children, you’re interested in the probabilities for the different possible numbers of girls in a family. Using theoretical probability (assume girls and boys are equally likely), compile a five-column table with the headings “0” through “4,” for the five possible numbers of girl children in a four-child family. Then, using “G” for girls and “B” for boys, list under each heading the various birth-order ways of achieving that number of girls in a family.

Then, use your table to calculate the following probabilities:

a. The probability of 1 girl b. The probability of 2 girls  c. The probability of 4 girls d. The probability the third child born is a girl

Answer:

0	1	2	3	4
1 BBBB P=1/16 = 0.0625 =6.25 %	4 GBBB BGBB BBGB BBBG P=4/16 = .25 =25 %	6 GGBB GBGB BGGB BBGG GBBG BGBG P=6/16 =0.375 =37.5 %	4 GGGB BGGG GGBG GBGG P=4/16 = .25 = 25 %	1 GGGG GGGG =0.0625 =6.25 %

2x2x2x2=16. GGGG, GGGB, GGBG, GBGG, BGGG, GGBB, GBGB, BGGB, BGBG, BBGG, GBBB, GBBB, BBGB, BBBG, BBBB, GBBG.

a. The probability of one girl: 25 %

b. The probability of two girls: 37.5 %

c. The probability of four girls: 6.25 %

d. Probability of the third child being a girl: 50 %

As pictured in Figure 6.11 of your textbook, a roulette wheel has 38 numbers: 18 odd black numbers from 1 to 35, 18 even red numbers from 2 to 36, and the two green numbers 0 and 00. Using theoretical probability, calculate the following:

a. The probability of spinning a green number b. The probability of spinning a number greater than 30 c. The probability of spinning a red number less than 10 d. The probability of spinning an even black number e. The expected total of green numbers in 57,000 spins

Answer:

a. The probability of spinning a green number: P= 2/38=0.05263=5.26 %

b. The probability of spinning a number greater than 30: (31, 32, 33, 34, 35, 36, 37, 38) P=8/38=0.2105=21.05 %

c. The probability of spinning a red number less than 10: (2, 4, 6, 8) P=4/38=0.1052=10.52 %

d. The probability of spinning an even black number: P=0/38=0.00=0 %

e. The expected total of green numbers in 57.000 spins: P (green numbers) 2/38 =1/19 E(X) = n x p

E(X) = 57.000 x 1/19

= 57000x 1/19

E(X)=3000

 In a nationwide polls of 1,500 randomly selected U.S. residents, 77% said that they liked pizza. In a poll of 1,500 randomly selected U.S. residents one month later, 75% responded that they liked pizza.

a. Does the polling evidence support the claim that pizza declined in popularity over the month between polls? Explain why or why not. b. Using statistical terminology, precisely identify the population parameter the two polls were attempting to measure. How does a parameter differ from a statistic? c. Based on the two polls, what would you say to someone who guessed that the population parameter the polls are trying to measure is really only 50%?

Answer:

n=1500= sample size

P1= first nationwide poll of U.S. Residents who like pizza = // % = 0.77

n1= 1500

P2 = second nationwide poll of U.S. Residents who like pizza, a month after the first sample: 75 % = 0.75

n2= 1500

Hº: P1=P2 H1: P1>P2

P= n1 P1+n2 P2/ (n1+n2)

P= (1500(0.77)) + (1500(0.75))/1500+1500)

P= 1155 + 1125/(30009

P= 2280/ (3000)

P= 0.76

z= P1-P2/ (√P(1-p)(1/n1)+(1/n2))z= 0.77-0.75/ (√0.76(1-0.76)((1/1500)+(1/1500))

z= 0.2/(√0.76 (1-0.76)(1/750)

z= 0.02/ (√0.1824(1/750)

z= (0.02)/√0.1824/5√30)

z=5√30 (0.02)/√0.1824

z=v√30(0.1)/√0.1824

z=0.1√5.472/0.1824

z= 1.2825

P-value = P (z > 1.2825)

P-value = 0.0998

P-value is > α

Since the p-value is greater than α, we reject H0 at 5 % level of significance.

a. The polling evidence indicates that pizza declined in popularity over one month. The H0 is rejected due to a greater p-value than α. Therefore, stating the H1: P1 >P2 is true, the first nationwide poll of people who like pizza is greater than the second poll a month later on people nationwide in the U.S. who like pizza. This suports the claim that pizza declined in popularity over the month between polls.

b. A population parameter is the number of the total population that the poll is mentioning. In this sample, 1500 people were chosen to determine the percentage of the amount of people who liked pizza. A parameter is a characteristic of a population. A statistic is a characteristic of a sample.

c. Sample is not enough to support.

Eleven people have eleven different favorite numbers from 2 to 12. They all agree to participate in a 10,000-roll dice game where they bet $1 on their favorite number for each roll of two standard (fair) dice. A donor kicks in an extra dollar every round, so the payoff if your number comes up is $12.

a. Assuming everyone bets on all 10,000 rounds, what is the expected value for a person who has number 7? (Show your calculations.) b. Assuming everyone bets on all 10,000 rounds, what is the expected value for a person who has number 2? (Show your calculations.)

Answer:

a. P(7)=((1,6)(2,5)(3,4)(4,3)(5,2)(6,1))/36 = 6/36 = 1/6

E(X) = n x p = 10000 x (12 x 1/6 – 1 x (1 – 1/6)) = 1000x (12 x 1/6 – 1 x 5/6)

E(X)= 11666.66

b. P(2)=(1,1)/36 = 1/36 E(X) = n x p = 10000 x (12 x 1/36 – 1 (1- 1/36)) = 10000 x (12 x 1/36 – 1 x 35/36)

E(X)= 6388.88

In the following situations, indicate whether you’d use the normal distribution, the t distribution, or neither.

a. The population is normally distributed, and you know the population standard deviation. b. You don’t know the population standard deviation, and the sample size is 35. c. The sample size is 22, and the population is normally distributed. d. The sample size is 12, and the population is not normally distributed. e. The sample size is 45, and you know the population standard deviation

Answer:

a. Since the population is normally distributed and we know the population standard deviation, it is a normal distribution (z-distribution)

b. We don´t know the population standard deviation, but we know the sample size is 35. We will do a regular t-distribution.

c. With a normal distribution and a sample size of 22, we will do a regular t-distribution.

d. Since the population isn´t normal and the sample size is 12, we will do neither distribution

e. Knowing the population standard deviation and the sample size, we will do a regular distribution (z-distribution)

The prices of used books at a large college bookstore are normally distributed. If a sample of 23 used books from this store has a mean price of $27.50 with a standard deviation of $6.75, use Table 10.1 in your textbook to calculate the following for a 95% confidence level about the population mean. Be sure to show your work.

a. Degrees of freedom b. The critical value of t c. The margin of error d. The confidence interval for a 95% confidence level

Answer:

Sample size: n:23

Sample mean: x̄ :27.50

Sample standard deviation: σ: 6.75

95 % confidence level

Level of significance: = α = 1-95 = 0.05

a. Degrees of freedom, df-n-1=23-1= 22 degrees of freedom

b. The critical value of t: 1-α/2, df=1-0.05/2, 22=0.975, 23 = 2.074

c. The margin of error: z x σ/√n= 1.96x6.75√23= (1.96 x 6.75√23)/23=(13.23√23)/23=2.758=2.76

d. x̄ ± z x σ/√n = 27.50±1.96x6.75/√23= 27.50+1.96x6.75/√23=27.50+(13.23√23)/23=30.2586 27.50-1.96x6.75√23=27.50-(13.23√23)/23= 24.7413

 Statistics students at a state college compiled the following two-way table from a sample of randomly selected students at their college:

	Play chess	Don’t play chess
Male students	25	162
Female students	19	148

Answer the following questions about the table. Be sure to show any calculations.

a. How many students in total were surveyed? b. How many of the students surveyed play chess?  c. What question about the population of students at the state college would this table attempt to answer? d. State Hº  and Hª for the test related to this table

Answer:

a. 25+19+162+148=354

b. 25+19=44

c. The population of male students who play chess (p1) is equal to the population of female students who play chess (p2).

Hº: p1=p2

Hª: p1≠p2

Answer the following questions about an ANOVA analysis involving three samples.

a. In this ANOVA analysis, what are we trying to determine about the three populations they’re taken from? b. State the null and alternate hypotheses for a three-sample ANOVA analysis.  c. What sample statistics must be known to conduct an ANOVA analysis? d. In an ANOVA test, what does an F test statistic lower than its critical value tell us about the three populations we’re examining?

Answer:

a. What we are trying to determine about the three populations that ANOVA is taken from is the difference in means. It´s a way to figure out if the survey or experiment results are significant.

b. Hº: μ1=μ2=μ3

Hª: At least one of the means is different from the others

c. Sample statistics of the sample size, mean and standard deviations of all comparison groups must be known in order to conduct an ANOVA analysis.

d. When the F test statistic is lower than it´s critical value, this tells us that at least one of the three populations means we are comparing is different. This indicates that the null hypothesis is rejected.