Question

1. 7.75 p. 428

Salaries for teachers in a particular elementary school district are normally distributed with a mean of $44,000 and a standard deviation of $6,500. We randomly survey ten teachers from that district.

a. In words, X = ______________

b. X ~ _____(_____,_____)

c. In words, ΣX = _____________

d. ΣX ~ _____(_____,_____)

e. Find the probability that the teachers earn a total of over $400,000.

f. Find the 90th percentile for an individual teacher's salary.

g. Find the 90th percentile for the sum of ten teachers' salary.
h. If we surveyed 70 teachers instead of ten, graphically, how would that change the distribution in part d?

i. If each of the 70 teachers received a $3,000 raise, graphically, how would that change the distribution in part b?

2. 7.90 p. 431

The average length of a maternity stay in a U.S. hospital is said to be 2.4 days with a standard deviation of 0.9 days. We randomly survey 80 women who recently bore children in a U.S. hospital.

a. In words, X = _____________

b. In words, X ¯ = ___________________

c. X ¯ ~ _____(_____,_____)

d. In words, ΣX = _______________

e. ΣX ~ _____(_____,_____)

f. Is it likely that an individual stayed more than five days in the hospital? Why or why not?

g. Is it likely that the average stay for the 80 women was more than five days? Why or why not? h. Which is more likely:

i. An individual stayed more than five days.

ii. the average stay of 80 women was more than five days.

i. If we were to sum up the women’s stays, is it likely that, collectively they spent more than a year in the hospital? Why or why not?

(3) Finding Dory Coral Reef communities are home to one-quarter of all marine plants and animals worldwide. These reef support large fisheries by providing breeding grounds and safe havens for young fish of many species. Coral reefs are seawalls (protecting shorelines from tides, storm-surges, and hurricanes as well as produce the limestone and sand of which beaches are made), Marine scientists say that a tenth of the world’s reef systems have been destroyed in recent times. At current rates of loss, almost three-quarters of the reefs could be gone in 30 years. A particular lab studies corals and the diseases that affect them. Dr. Drew Harvell and his lab sampled sea fans at 19 randomly selected reefs along the Yucatan peninsula and diagnosed whether the animals (the sea fans) were affected by aspergillosis1 . In specimens collected at a depth of 40 feet at the Las Redes Reef in Akumal, Mexico, scientists found that 52% of the 104 sampled sea fans were infected with aspergillosis.

(a) What are the mean (proportion, p) and standard deviation of the sampling distribution of the sample proportion (mean (p) and sepˆ) of infected sea fans? What should the distribution look like (think of the definition of CLT)?

(b) What is probability that the sample proportion of infected sea fans is less than 50% (that is find P(ˆp < .5))?

(c) What is probability that the sample proportion of infected sea fans is between 40 and 60%?

(4) There is no Dana, only Zeul (Who you gonna call?) In November of 2005 the Harris Poll asked 889 randomly selected US adults, “Do you believe in ghosts?” 29% said they did.

(a) In constructing confidence intervals, would we use z ? or t ? in this situation? Briefly explain why you would use one instead of the other.

(b) Estimate p, the true proportion of US adults that believe in ghosts, with 90% confidence. Interpret the interval in context of the data.

(c) Suppose, using the information from the survey (the 29% that believe in ghosts) that a new survey is to be taken and the new bound is to be 2%. What sample size will be required?

(d) Suppose that we know nothing of any prior results from this study (thus have no estimate for the proportion of those US adults that believe in ghosts). What proportion should we use for the estimation? What sample size do we need with no prior information? Why is it different than the sample size from part (c).

(5) Got Milk? Although most of us buy milk by the quart or gallon, farmers (at least in the US) measure daily production in pounds (lbs.). Ayrshire cows have a known standard deviation of 6 pounds and average 47 pounds of milk per day. Jersey cows have a known standard deviation of 5 pounds and the mean daily production is 43 pounds. Assume that the distribution of daily milk production is approximately normal and suppose one farm has 20 of each type of cow (20 Ayrshire and 20 Jersey).

(a) In constructing confidence intervals, would we use z ? or t ? in this situation? Briefly explain why you would use one instead of the other.

(b) Estimate µ, the true mean daily milk production of both Ayrshire and Jersey cows (you will have 2 CIs), with 95% confidence. Interpret.

(c) Suppose the next time the farmer takes a sample of Ayrshire cows, he wants to make sure the bound is 2. Maintaining 95% confidence, what sample size will be required for the new sample?

(6) Using the t table Find the degrees of freedom (df) and the value of t ? for the given sample size and confidence level or significance level (α). [Hint: if it states ‘CL’, that means that α is divided by 2. If it says ‘α = ’, then you do not divide α by 2.]

(a) n = 6,CL = 90%

(b) n = 21,CL = 98%

(c) n = 29,CL = 95%

(d) n = 12,CL = 99%

(e) n = 6,α = 0.10

(f) n = 21,α = 0.01

(g) n = 40,α = 0.05

(7) It ain’t easy bein’ green A dealer in recycled paper places empty trailers at various sites. The trailers are gradually filled by individuals who bring in old newspapers and magazines, and are picked up on several schedules. One such schedule involves pickup every second week. This schedule is desirable if the average amount of recycled paper is more than 1600 cubic feet per 2-week period. Below is a copy of the dealer’s records for eighteen 2-week periods show the following volumes (in cubic feet) at a particular site; the mean and standard deviation are as follows: X¯ = 1721.6 and s = 154.5

(a) In constructing confidence intervals, would we use z ? or t ? in this situation? Briefly explain why you would use one instead of the other.

(b) Estimate the true mean weight of recycled paper with 95% confidence. Interpret.

recycle=c(1935,1556,1752,1969,1804,1842,1994,1810,1827,1725,2003,1499,1809,1795,1622,1620,1777,2035)

(8) 9.7 p. 535

In a population of fish, approximately 42% are female. A test is conducted to see if, in fact, the proportion is less. State the null and alternative hypotheses

(9) 9.9 p. 535

A random survey of 75 death row inmates revealed that the mean length of time on death row is 17.4 years with a standard deviation of 6.3 years. If you were conducting a hypothesis test to determine if the population mean time on death row could likely be 15 years, what would the null and alternative hypotheses be?

a. H_0: __________

b. H_a: __________

(10) 9.62 p. 538

Some of the following statements refer to the null hypothesis, some to the alternate hypothesis. State the null hypothesis, H0, and the alternative hypothesis. Ha, in terms of the appropriate parameter (μ or p).

a. The mean number of years Americans work before retiring is 34.

b. At most 60% of Americans vote in presidential elections.

c. The mean starting salary for San Jose State University graduates is at least $100,000 per year.

d. Twenty-nine percent of high school seniors get drunk each month.

e. Fewer than 5% of adults ride the bus to work in Los Angeles.

f. The mean number of cars a person owns in her lifetime is not more than ten.

g. About half of Americans prefer to live away from cities, given the choice.

h. Europeans have a mean paid vacation each year of six weeks.

i. The chance of developing breast cancer is under 11% for women.

j. Private universities' mean tuition cost is more than $20,000 per year

Answer 1

1. 7.75 p. 428

a. In words, X=X is random variable that denotes Salaries for teachers in a particular elementary school district

b. X ~ N(µ , σ)

X ~ N(44000,6500)

c. In words, ΣX =denotes total salary of ten teachers in a particular elementary school district

d. ΣX ~ N(nµx, √n σx)

~N(440000,20554.80)

e. Find the probability that the teachers earn a total of over $400,000

µ =    440000                          
σ =    20554.80   
right tailed                              
P ( X ≥   400000   )                      
                              
Z =   (X - µ ) / σ = (   400000.00   -   440000   ) /    20554.80 =   -1.946
                              
P(X ≥   400000.000   ) = P(Z ≥   -1.946   ) =   P ( Z <   1.946   ) =    0.9742(answer)

excel formula for probability from z score is =NORMSDIST(Z)                              

.

f. Find the 90th percentile for an individual teacher's salary.

µ =    44000                  
σ =    6500                  
proportion=   0.90                  
                      
Z value at    0.9   =   1.2816   (excel formula =NORMSINV(   0.90   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   1.282   *   6500   +   44000  
X   =   52330

g. Find the 90th percentile for the sum of ten teachers' salary.

µ =    440000                  
σ =    20554.80
proportion=   0.90                  
                      
Z value at    0.9   =   1.2816   (excel formula =NORMSINV(   0.90   ) )
z=(x-µ)/σ                      
so, X=zσ+µ=   1.282   *   20554.80 +   440000  
X   =   466342.04