**Self Test**** 6 for Probability and
Statistics**

This test was constructed by Zauresh
Atakhanova based on the Schaum's
Outline *Theory and Problems of Probability and Statistics* by Murray R.
Spiegel with contributions from others as noted below. If you need more review
refer also to this outline.

**1. True False** Give an example of estimators (or estimates)
which are (a) unbiased and efficient, (b) unbiased and inefficient, (c) biased
and inefficient.

a) sample mean

b) sample median

c) sample standard deviation

**Confidence Interval Estimates for Means (Large Samples)**

**2.****True
False**. Measurements of the diameters of a
random sample of 196 ball bearings made by a certain machine during one week
showed a mean of 0.724 inches and a standard deviation of 0.042. Find (a) 95%
and (b) 99% confidence interval for the mean diameter of all the ball bearings.

a) 0.724+/-0.0077

b) 0.724+/-0.0058

**Confidence Interval Estimates for Means (Small Samples)**

**3.****True
False.**
The 95% confidence coefficients (1.96 'two-tailed') for the normal distribution
are given by +/-1.96. What are the corresponding coefficients for the t
distribution if (a) v=9, (b) v=30?

(a) +/-2.26

(b) +/-2.04

**4.****True
False.** A sample poll of 1000 voters chosen at
random from all voters in a given district indicated that 45% of them were in
favor of a particular candidate. Find (a) 95%, (b) 99% and (c) 99.73%
confidence limits for the proportion of all the voters in favor of this candidate.

(a) 0.45+/-0.0106

(b) 0.45+/-0.0157

(c) 0.45+/-0.0472

**5.****True
False.** A sample of 150 brand
A light bulbs showed a mean lifetime of 1500 hours and a standard deviation of
1300 hours. A sample of 200 brand B light bulbs showed a mean lifetime of 1200
hours and a standard deviation of 80 hours. Find (a) 95% and (b) 99% confidence
limits for the difference of the mean lifetimes of the populations of brands A
and B.

a) 200+/-20.8

b) 200+/-28.6

**6.****True
False.**
The standard deviation of the lifetimes of a sample of 200 electric light bulbs
was computed to be 150 hours. Find (a) 95% and (b) 99% confidence intervals for
the standard deviation of all such light bulbs.

a) 150+/-5.3

b) 150+/-8.7

**7.****True
False**.Work problem 5 using small sampling theory.

(a) (136.86, 167.03)

(b) (133.42, 173.88)

**8.****True
False**. Two samples of sizes 15 and 10 are drawn
at random from two normal populations. If their variances are found to be 24
and 16 respectively, with 90% confidence interval for the ratio of the
variances is (0.4804, 3.7451)

**9. True
False **A copper smelter in northern

**More Confidence Interval Estimates for Means (Large
Samples)**

**10. True
False **After
extensive test drilling of on area of possible gold ore, a random sample of 200
Kg. Drill core were measured by a machine to find their gold content. The
sampling showed a mean of 8.24g of gold per sample with a standard deviation of
0.42g. Find (a) 95% and (b) 99% confidence interval for the mean gold content
of all the samples.

a) 8.24 __+__ 0.058g

b) 8.24 __+__ 0.77g

**11. True
False **A
sample of 200 employees has found that the average number of days off sick in a
year was 8.2, with a standard deviation of 3.5. Then the average number of days
off sick to be for the employed population is the 95% confidence interval for which is** ****. **(Contributed by Mansoor
Al-Harthy)

**12. True
False **Currently,
environmental regulations allow your factory to have maximum indoor emissions
of particular matter (PM-2.5) of an annual mean concentration of 40 mg/m^{3}
(arithmetic mean). Local inspectors collected measurements from your
plant 18 times last year; these measurements resulted in a mean concentration
of PM-2.5 of 45 mg/m^{3} with a
sample standard deviation of 8. Assuming that the concentration of PM-2.5 from
your plant follows a normal distribution, the inspectors correctly concluded
with 99% confidence that your plant last year was in violation of the
environmental regulations and you should be fined accordingly. (Contributed
by Alex Lombardia)