Statistics 401B
Fall 2014
Laboratory Assignment 5
1. An Izod impact test (an ASTM standard method of determining the impact resistance of materials) was
performed on 20 specimens of PVC pipe. The sample average and standard deviation obtained were
ȳ = 1.113 and s = 0.325, respectively. Use hand computation to do all parts of this problem.
(a) The ASTM standard for this material requires that the mean Izod impact strength must be greater
than 1.0 ft-lb/in. Is there sufficient evidence that the mean Izod impact strength µ, exceeds 1.0 ftlb/in? State the null and alternative hypotheses and perform the test using α = .05.
(b) Assuming that the standard deviation of the loads at specimen failure is σ ≈ 0.5ft-lb/in , use the
charts in Table 3 to determine (to the best accuracy possible) the power of the above test to detect a
difference when the true mean Izod impact strengths are 1.1,1.2,1.3 and 1.4 respectively.
µa
1.1
d
β
Power=1-β
1.2
1.3
1.4
(c) Explain why the following are reasonable assessments based on the solution to part (b).
i. There is a very good chance of falsely failing to reject the null hypothesis (false negative) if the
mean Izod impact strength actually is larger by only 0.1 ft-lb/in over the current standard.
ii. There is a very good chance of rejecting the null hypothesis (i.e., detecting a difference) if the
mean Izod impact strength actually is as large as 1.4 ft-lb/in over the current standard.
(d) Before the tests were run, suppose that you were required to make a sample size determination to
ensure that the Type II error probability of accepting H0 is kept below 0.1 when the actual mean
Izod impact strength exceeds 1.15 ft-lb/in, if α = .05 is used for a right-tail test and σ ≈ 0.3. Obtain
an approximate value for the minimum sample size necessary using the sample size table supplied in
class (or the sample size calculator demonstrated in class).
1
2. An industrial concern has experimented with several different mixtures of the four components - magnesiun, sodium nitrate, strontium nitrate, and a binder- that comprise a rocket propellant. The company has
found that a new blend may give higher flare-illumination values than the standard. The standard blend,
Mixture 1, is composed of the components in the proportions .40, .10, .42, and .08, respectively; Mixture
2, the new blend, uses the proportions .60, .27, .10, and .05. Twenty different samples (10 of each mixture)
were prepared independently and tested to obtain the following flare-illumination values (in units of 1,000
candles):
Mixture
y, Flare Illumination(in 1,000 candles)
y
y2
1
185 192 201 215 170 190 175 172 198 202 1900 362,932
2
221 210 215 202 204 196 225 230 214 217 2134 456,412
Note: Use hand calculation for parts (a) to (d) and show work. Use an appropriate JMP analysis of the
data for answering parts (e) to (g). Turn in the marked JMP output and show work.
(a) Calculate the sample means ȳ1 , ȳ2 , the sample variances s21 , s22 , and the pooled standard deviation
sp by hand, using a calculator and the summaries given above.
P
P
(b) Do the data provide sufficient evidence that Mixture 2 has a higher flare-illumination than that of Mixture 1? Specify appropriate hypotheses tested in terms of the difference in the mean flare-illumination
in the two populations, µ1 and µ2 , respectively, calculate the value of the appropriate test statistic,
and make a decision using α = .05.
(c) Calculate a 90% confidence interval on the difference in the mean flare-illumination in the two populations. Use this interval to test the hypothesis in part (b) stating the α level at which the hypothesis
is tested.
(d) Suppose that before the experiment the investgators wished to determine a sample size appropriate for
testing the research hypothesis that µ1 < µ2 . Assuming that the populations have normal distributions
with common standard deviation σ = 12, calculate the sample size required to obtain a test with
β < .10 when µ1 − µ2 ≤ 15 if an α = .05 is used for the test? Must show work.
2
(e) Assess the validity of the three conditions necessary for the test used in part (b) to be valid. Give
reasons for assuming that these are satisfied or not to an acceptable degree?
(f) Circle the t-statistic for testing the hypothesis you stated in Part (b) and the p-value associated with
it in the JMP output. Extract them from the JMP output and write them here also.
(g) Extract the 90% confidence interval for the difference in population means as in Part (d) from the
JMP output. Write it here and circle it in the JMP output.
3
3. Arsenic concentration in public drinking water supplies is a potential health hazard. An article in the
Arizona Republic(Sunday May 27, 2001) reported drinking water concentrations in parts per billion (ppb)
for 10 metropolitan Phoenix communities and 10 communities in rural Arizona. The data follows:
Metro
Rural
Phoenix, 3
Rimrock, 48
Chandler, 7
Goodyear, 44
Gilbert, 25
New River, 40
Glendale, 10
Apache Junction, 38
Mesa, 15
Buckeye, 33
Paradise Valley, 6 Nogales, 21
Peoria, 12
Black Canyon City, 20
Scottsdale, 25
Sedona, 12
Tempe, 15
Payson, 1
Sun City, 7
Casa Grande, 18
P
y=
125
275
P 2
y = 2087
9683
Note: Use hand calculation for parts (a) to (c) and show work. Use an appropriate JMP analysis of the
data for answering parts (d),(e), and (f). Turn in the marked JMP output and show work.
(a) Calculate the sample means ȳmetro and ȳrural and sample variances s2metro and s2rural , respectively, of
the two samples. Is there evidence in these numbers to suspect that the variances of the two populations of beams may be different? Explain.
(b) We wish to determine if there is any difference in the mean arsenic concentrations in the public
drinking water between metropolitan Phoenix and rural Arizona. First, state H0 and Ha in terms of
appropriate population parameters you define. Use the approximate two sample t-test (Welch’s test)
applicable for independent samples from populations with unequal variances and α = .05.
4
(c) Compute a 95% confidence interval for the difference in population means of arsenic concentrations
between metropolitan Phoenix and rural Arizona assuming unequal population variances.
(d) Find evidence in your JMP output (a) for supporting the normality assumption of the two samples,
and (b) for supporting the supposition that the population variances are not equal.
(e) The t-statistic for testing the hypothesis you stated in Part (b) under the unequal variances assumption can be computed in JMP. Circle it in the JMP output and copy here the statistic and the p-value
associated with it extracted from the JMP output.
(f) Extract the 95% confidence interval for the difference in population means as in Part (c) computed
under unequal variances assumption from the JMP output. Copy it here and circle it in the JMP
output.
Due Thursday, October 23rd, 2014 (turn-in during the first 20 min. of the lab)
5