Download Name___________________ STA 6166 Exam #1 Fall 2002 1. pH

Name___________________
STA 6166
Exam #1
Fall 2002
1. pH values in soil cores from a county are thought to be normally distributed with a mean of
6.5 and a standard deviation of 0.5.
(8) a. What is the probability that the pH of one randomly selected soil core will exceed 6.6?
P{y>6.6} = P{(y-6.5)/0.5 > (6.6-6.5)/0.5} = P{z>.2} = .421
(8) b. What is the probability that the average pH of 25 randomly drawn cores will exceed 6.6?
Note: Sampling distribution of ybar is normal, mean = 6.5 and std dev = 0.5/√25 = 0.1
P{ybar>6.6} = P{(ybar-6.5)/0.1 > (6.6-6.5/0.1} = P{z>1} = 0.159
(8) c. You now discover that pH values in the county are not normally distributed, but the mean
is 6.5 and the standard deviation is 0.5. Which of your two answers (to a. and b. above) do you
continue to believe is true, at least approximately so? What statistical principle justifies this
belief? Explain.
Answer to b. is more likely to be correct than answer to a. because the central limit theorem
applies to sampling distribution of ybar.
2. A mining land reclamation site is being evaluated for approval by a regulatory agency. In
order for the site to be approved, the mean plant coverage at the site must be at least 80%.
Percent plant cover was measured in 16 randomly chosen areas (10 m sq) at the site.
(8) a. What is the population to which you wish to make inference? What is a symbol for the
mean of that population?
The population is the set of coverage percentages of all 10x10 m square areas in the entire site.
The symbol for the mean of that population is µ
(8) b. What is the sample from which you will obtain information to make the inference in a?
What must be true about this sample for the inference to be valid? What is a symbol for the mean
of that sample?
The sample is the set of coverage percentages in the 16 10x10 m squares that were actually
measured.
The symbol for the mean of the sample is ybar.
(8) c. The mean of the 16 values in the sample was 82%. Construct a 95% confidence interval
for the mean coverage. Assume the population standard deviation is σ = 8%. Are you convinced
that the reclamation site meets the agency criterion? Why or why not?
95% confidence interval: ybar ± t.025 s/√n
ybar = 82, s = 8, n= 16
95% confidence interval: 82 ± 2.13 (8/4) = 82 ± 2.13 (8/4) = 82 ± 4.26
No, because CI contains values less than 80.
3. Defective bolts have been accidentally mixed with good bolts giving a mix of 10% defective
and 90% good. Four bolts are drawn at random.
(8) a. What is the probability that all four are good?
P{all four good} = .94 =.656
(8) b. What is the probability that at least two are defective?
P{at least two defective}
= P{2, 3, or 4 defective}
= 1 - P{0 or 1 defective}
= 1 – P{0 defective} – P{1 defective}
= 1 – .94 – 4[(.1)(.9)3]
= 1 - .656 - .292
4. Average weight of 12-year old children in 1980 was 85 lbs. You have heard that children are
heavier now than in 1980, so you conduct a study to see if this is true. You measure weights of
fifty 12-year old children and find a mean of 87 and standard deviation of 15.
(8) a. Construct a test statistic for testing H0:µ=85 versus Ha:µ>85.
Test statistic t = (ybar – 85)/(15/√50) = (87 – 85)/(15/√50) = .94
(8) b. Determine the significance level of your test.
p-value = .176
5.
Short answer and true-false (2 each):
a. (T or F) The number of kittens in a litter is a discrete random variable. _Y___
b. The interval between the 25th and 75th percentiles is called the ____inter-quartile
range________.
c. Standard deviation and range are measures of __dispersion__ in a set of data.
d. Name two graphical techniques to describe data. __boxplot_____________ and
__histogram__________.
e. The median of the set of numbers 3, 7, 4, 3, 8 is _4___.
f. The mode of the set of numbers in g. is ___3_.
g. The range of the set of numbers in g. is __5__.
h. The mean of the sampling distribution of a sample mean is ______µ____________.
i. The variance of the sampling distribution of a sample mean is _____σ2/n__________.
j. I’m a real Gator fan, even in times of doubt __Unfailingly_.