## [answered] MACT 3224/317: Assignment N 5 Yahia El Horbaty, Khouzeima M

MACT 3224/317: Assignment N 5

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? Late submissions are not allowed. ? The Department of Mathematics and Actuarial Science, The American University in Cairo. 1 ? Question 1:

An investigator computes a 95% confidence interval for a population mean on the basis of a sample of

size 70. If she wishes to compute a 95% confidence interval that is half as wide, how large a sample

does she need?

? Question 2:

A random sample of the number of barrels (in millions) of oil produced per day by world oil producing

countries is listed here:

3.56 1.90 7.83 2.83 1.91 5.88 2.91 6.08 (a) Find the sample mean and the sample standard deviation.

(b) Estimate the mean oil production per day with 95% confidence.

? Question 3:

An electrical engineer wants to compare the mean melting points for two di?erent metal alloys used

in soldering. He collects a sample of size n1 = 36 for the melting point of alloy 1 and finds a sample

average 185 o C. He collects a sample of size n2 = 64 for melting point of alloy 2 and finds a sample

average of 183 o C. Assume that the population standard deviation for the melting point of the first

alloy is 1 = 3 o C and the population standard deviation for the melting point of the second alloy is

o

2 = 4 C.

(a) Compute a 95% confidence interval for the di?erence of the means ?1

(b) Using a hypothesis test at level 0.05, is there evidence to claim that ?1 ?2 .

?2 = 1o C? ? Question 4:

A sample of 64 resistors are taken from a production line and are found to have a sample mean of

998 Ohms. The population distribution for the resistances is unknown but the population standard

deviation is known to be 10 Ohms.

(a) Find a 98% confidence interval for the population mean.

(b) Answer the following two questions by writing either ?narrower? or ?wider? in the blank space and

1. If the sample size were increased to n = 1000 then the confidence interval would be

the confidence interval computed from a sample with n = 64.

2. A 99% confidence interval is

than a 98% confidence interval. than ? Question 5:

A manufacturer claims that the average life of their electrical light bulbs is 2000 hours. A random

sample of 64 bulbs is tested and the life,x, in hours is recorded. The results obtained are as follows

Pn

?

i=1 xi = 127808

Pn

?

x)2 = 9694.6

i=1 (xi

(a) Is there sufficient evidence, at the 1%, that supports the manufacturer?s claims. (b) Is there sufficient evidence, at the 1%, that the manufacturer overestimates the length of life of

the light bulb.

? Question 6:

In communications let p denote the probability that a bit sent by the transmitter will be received erroneously on the other end of the channel. A telecom company has determined that for a communication

channel an error rate of more than 0.05 is unacceptable. To perform the test, a predetermined 32 bit

long message is sent (this is a sample with n = 32) and the number of erroneous bits are counted at

the receiving end. The number of erroneous bits is 2.

2 (a) Find a 99% confidence interval for the true probability p.

(b) Using a hypothesis test at level 0.01, can we conclude that the data present sufficient evidence to

suggest that the rate p is unacceptable?

? Question 7:

A questionnaire of spending habits was given to a random sample of college students. Each student

was asked to record and report the amount of money they spent on textbooks in a semester. The

sample of 130 students resulted in an average of \$422 with standard deviation of \$57.

1. Give a 90% confidence interval for the mean amount of money spent by college students on

textbooks.

2. How many students should you sample if you want a margin of error of \$5 for a 90% confidence

interval? 3

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