1

Supposexhas a distribution with a mean of90and a standard deviation of36. Random samples of size

n=64

are drawn.(a) Describe the

xdistribution

and compute the mean and standard deviation of the distribution.

x

has ---Select--- a binomial a normal an approximately normal a geometric a Poisson an unknown distribution with

mean_x=

and

standard deviation_x= .

(b) Find thezvalue corresponding to

x=94.5.

z=

(c) Find

P(x<94.5).

(Round your answer to four decimal places.)

P(x<94.5) =

(d) Would it be unusual for a random sample of size64from thexdistribution to have a sample mean less than94.5? Explain. Yes, it would be unusual because more than 5% of all such samples have means less than 94.5.No, it would not be unusual because more than 5% of all such samples have means less than 94.5. Yes, it would be unusual because less than 5% of all such samples have means less than 94.5.No, it would not be unusual because less than 5% of all such samples have means less than 94.5.

--

2

This question has several parts that must be completed sequentially. If you skip a part of the question, you will not receive any points for the skipped part, and you will not be able to come back to the skipped part.

Tutorial Exercise

Consider anxdistributionwith standard deviation=54.(a) If specifications for a research project require the standard error of the correspondingxdistributionto be9, how large does the sample size need to be? (b) If specifications for a research project require the standard error of the correspondingxdistributionto be 1, how large does the sample size need to be?

Step 1

(a) If specifications for a research project require the standard error of the correspondingxdistributionto be9, how large does the sample size need to be?

Recall that the standard deviation (also known as the standard error) of the

xdistribution

is

_x=

n

whereis the standard deviation of thexdistribution. Find the sample sizenfor the research project given that

_x=9and=54.

_x

=

n

=

54

n

n

=

n

=

3

Supposexhas a distribution with=30and=24.(a) If a random sample of sizen=34is drawn, find_x,_xandP(30x32). (Round_xto two decimal places and the probability to four decimal places.)

_x=

_x=

P(30x32) =

(b) If a random sample of sizen=64is drawn, find_x,_xandP(30x32). (Round_xto two decimal places and the probability to four decimal places.)

_x=

_x=

P(30x32) =

(c) Why should you expect the probability of part (b) to be higher than that of part (a)? (Hint: Consider the standard deviations in parts (a) and (b).) The standard deviation of part (b) is ---Select--- larger than smaller than the same as part (a) because of the ---Select--- same smaller larger sample size. Therefore, the distribution about_xis ---Select--- narrower wider the same .

4

Coal is carried from a mine in West Virginia to a power plant in New York in hopper cars on a long train. The automatic hopper car loader is set to put60tons of coal into each car. The actual weights of coal loaded into each car arenormally distributed, with mean=60tons and standard deviation=1.5ton.(a) What is the probability that one car chosen at random will have less than59.5tons of coal? (Round your answer to four decimal places.) (b) What is the probability that42cars chosen at random will have a mean load weightxof less than59.5tons of coal? (Round your answer to four decimal places.) (c) Suppose the weight of coal in one car was less than59.5tons. Would that fact make you suspect that the loader had slipped out of adjustment?YesNo Suppose the weight of coal in42cars selected at random had an averagexof less than59.5tons. Would that fact make you suspect that the loader had slipped out of adjustment? Why?Yes, the probability that this deviation is random is very small.Yes, the probability that this deviation is random is very large. No, the probability that this deviation is random is very small.No, the probability that this deviation is random is very large.

5

Suppose the heights of 18-year-old men are approximatelynormally distributed, with mean71inches and standard deviation5inches.(a) What is the probability that an 18-year-old man selected at random is between70and72inches tall? (Round your answer to four decimal places.) (b) If a random sample ofseventeen18-year-old men is selected, what is the probability that the mean heightxis between70and72inches? (Round your answer to four decimal places.) (c) Compare your answers to parts (a) and (b). Is the probability in part (b) much higher? Why would you expect this?The probability in part (b) is much lower because the standard deviation is smaller for thexdistribution.The probability in part (b) is much higher because the mean is larger for thexdistribution. The probability in part (b) is much higher because the standard deviation is larger for thexdistribution.The probability in part (b) is much higher because the standard deviation is smaller for thexdistribution.The probability in part (b) is much higher because the mean is smaller for thexdistribution.

6

Letxbe a random variable that represents the level of glucose in the blood (milligrams per deciliter of blood) after a 12 hour fast. Assume that for people under 50 years old,xhas a distribution that is approximately normal, with mean=82and estimated standard deviation=36. A test resultx<40 is an indication of severe excess insulin, and medication usually prescribed.(a) what the probability that, on a single test,x<40? (Round your answer to four decimal places.) (b) Suppose a doctor uses the averagexfor two tests taken about a week apart. What can we say about the probability distribution ofx?Hint: See Theorem 6.1.The probability distribution ofxis approximately normal with_x= 82 and_x= 18.00.The probability distribution ofxis approximately normal with_x= 82 and_x= 36. The probability distribution ofxis approximately normal with_x= 82 and_x= 25.46.The probability distribution ofxis not normal. What is the probability thatx<40? (Round your answer to four decimal places.) (c) Repeat part (b) forn= 3 tests taken a week apart. (Round your answer to four decimal places.) (d) Repeat part (b) forn= 5 tests taken a week apart. (Round your answer to four decimal places.) (e) Compare your answers to parts (a), (b), (c), and (d). Did the probabilities decrease asnincreased?YesNo Explain what this might imply if you were a doctor or a nurse.The more tests a patient completes, the weaker is the evidence for excess insulin.The more tests a patient completes, the weaker is the evidence for lack of insulin. The more tests a patient completes, the stronger is the evidence for lack of insulin.The more tests a patient completes, the stronger is the evidence for excess insulin.

7

Letxbe a random variable that represents white blood cell count per cubic milliliter of whole blood. Assume thatxhas a distribution that is approximately normal, with mean=7050and estimated standard deviation=2200.A test result ofx<3500is an indication of leukopenia. this indicates bone marrow depression that may be the result a viral infection.(a) what is probability that, on single test,xis less than 3500? (Round your answer to four decimal places.) (b) Suppose a doctor uses the averagexfor two tests taken about a week apart. What can we say about the probability distribution ofx?The probability distribution ofxis approximately normal with_x= 7050 and_x= 1555.63.The probability distribution ofxis approximately normal with_x= 7050 and_x= 1100.00. The probability distribution ofxis approximately normal with_x= 7050 and_x= 2200.The probability distribution ofxis not normal. What is the probability ofx<3500? (Round your answer to four decimal places.) (c) Repeat part (b) forn= 3 tests taken a week apart. (Round your answer to four decimal places.) (d) Compare your answers to parts (a), (b), and (c). How did the probabilities change asnincreased?The probabilities decreased asnincreased.The probabilities stayed the same asnincreased. The probabilities increased asnincreased. If a person hadx<3500 based on three tests, what conclusion would you draw as a doctor or a nurse?It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia.It would be an extremely rare event for a person to have two or three tests below 3,500 purely by chance. The person probably does not have leukopenia. It would be a common event for a person to have two or three tests below 3,500 purely by chance. The person probably does not have leukopenia.It would be a common event for a person to have two or three tests below 3,500 purely by chance. The person probably has leukopenia.

8

A European growth mutual fund specializes in stocks from the British Isles, continental Europe, and Scandinavia. The fund has over425stocks. Letxbe a random variable that represents the monthly percentage return for this fund. Supposexhas mean=1.7%and standard deviation=0.7%.(a) Let's consider the monthly return of the stocks in the fund to be a sample from the population of monthly returns of all European stocks. Is it reasonable to assume thatx(the average monthly return on the425stocks in the fund) has a distribution that is approximately normal? Explain. ---Select--- Yes No ,xis a mean of a sample ofn=425stocks. By the ---Select--- theory of normality central limit theorem law of large numbers , thexdistribution ---Select--- is is not approximately normal. (b) After 9 months, what is the probability that theaveragemonthly percentage returnxwill be between 1% and 2%? (Round your answer to four decimal places.) (c) After 18 months, what is the probability that theaveragemonthly percentage returnxwill be between 1% and 2%? (Round your answer to four decimal places.) (d) Compare your answers to parts (b) and (c). Did the probability increase asn(number of months) increased? Why would this happen?No, the probability stayed the same.Yes, probability increases as the standard deviation decreases. Yes, probability increases as the mean increases.Yes, probability increases as the standard deviation increases. (e) If after 18 months the average monthly percentage returnxis more than 2%, would that tend to shake your confidence in the statement that=1.7%?If this happened, do you think the European stock market might be heating up? (Round your answer to four decimal places.) P(x> 2%) = Explain. This is very likely if= 1.7%. One would suspect that the European stock market may be heating up.This is very likely if= 1.7%. One would not suspect that the European stock market may be heating up. This is very unlikely if= 1.7%. One would suspect that the European stock market may be heating up.This is very unlikely if= 1.7%. One would not suspect that the European stock market may be heating up.

9

The taxi and takeoff time for commercial jets is a random variablexwith a mean of8.3minutes and a standard deviation of3.4minutes. Assume that the distribution of taxi and takeoff times is approximately normal. You may assume that the jets are lined up on a runway so that one taxies and takes off immediately after the other, and that they take off one at a time on a given runway.(a) What is the probability that for37jets on a given runway, total taxi and takeoff time will be less than 320 minutes? (Round your answer to four decimal places.) (b) What is the probability that for37jets on a given runway, total taxi and takeoff time will be more than 275 minutes? (Round your answer to four decimal places.) (c) What is the probability that for37jets on a given runway, total taxi and takeoff time will be between 275 and 320 minutes? (Round your answer to four decimal places.)

(hw 6.5)