STATISTICS 402B
Spring 2016
Homework Set#1
1. Problem 2.6
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
1
2
185
221
y, Flare Illumination(in 1,000 candles)
192 201 215 170 190 175 172 198
210 215 202 204 196 225 230 214
P
202
217
y
1900
2134
P 2
y
362,932
456,412
Note: Use hand calculation for parts (a) to (d) and show work. Use an appropriate JMP analysis of the
JMP data table flare.jmp for answering parts (e) to (g). Turn in the marked JMP output and show
work.
(a) Calculate the sample means y¯1 , y¯2 , the sample variances s21 , s22 , and the pooled standard deviation
sp by hand, using a calculator and the summaries given above.
(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 flareillumination in the two populations, µ1 and µ2 , respectively, calculate the value of the appropriate
test statistic, and make a decision using α = .05 and the t-table.
(c) Calculate a 90% confidence interval on the difference in the mean flare-illumination µ1 − µ2 of 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 (i.e., H1 ) that µ1 < µ2 . Assuming that the populations have normal
distributions with common standard deviation σ = 10, 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? Use the β curves supplied, but
may verify your answer using an internet calculator. Must show work.
(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? (You may use graphical
and statistical tools available from the JMP output.)
(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. A new alloy is proposed for the manufacture of steel beams. The company’s metallurgist designs a study
to compare the strength of the new alloy to the currently used alloy. Ten beams of each type of alloy are
manufactured. The load capacities (in tons) for the 20 beams are determined and given below:
Alloy
Old
New
23.7
26.6
24.0
32.3
21.1
28.0
y, Load Capacity (tons)
23.1 22.8 25.0 25.3
31.1 29.6 28.5 31.2
1
P
22.6
23.6
23.3
29.1
22.8
28.5
y
233.7
288.5
P 2
y
5474.93
8379.93
Note: Use hand calculation for parts (a) to (c) and show work. Use an appropriate JMP analysis of the of
the JMP data table beam.jmp for answering parts (d,(e), and (f). Turn in the marked JMP output and
show work.
(a) Calculate the sample means ȳold and ȳnew and sample variances s2old and s2new , respectively, of the
two samples. Is there evidence to believe that the variances of the two populations of beams may be
different? Explain.
(b) Test the research hypothesis that the mean load capacity of beams made with the new alloy is greater
than that of beams made with the old alloy. 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 = .01.
(c) Compute a 98% confidence interval for the difference in population means of beams made of the two
alloys 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 state the statistic and the p-value associated
with it extracted from the JMP output in your answer sheet.
(f) Extract the 98% confidence interval for the difference in population means as in Part (c) computed under unequal variances assumption from the JMP output. state the statistic and the p-value associated
with it extracted from the JMP output in your answer sheet and circle it in the JMP output.
4. Problem 2.24 (Must be carried out using hand calculations entirely; do not use JMP.)
5. Problem 2.29 (Use JMP output to provide answers; use the JMP data table 2-29.jmp; use JMP power and
sample size calculator)
6. In the Portland cement experiment in Chapter 2, use the OC curves supplied to find the approximate
sample size needed to be able to detect a difference in mean strengths of at least 0.90 kgf /cm2 using
α = .05, with a power of at least 0.9. Assume that from historical data, an estimate of σ is 0.5 and that
n1 = n2 = n i.e., equal sample sizes are expected to be obtained from both modified and unmodified
mortar. Show all work. (You may use the power calculator from JMP to verify your answer)
7. Problem 2.34 (Use JMP output to provide answers; use the JMP data table 2-34.jmp)
Note: You may use software (say, Excel to compute elementary statistics such as means and variances) if needed.
Attach computer output when necessary as proof of your calculations or for producing any graphics needed. Also
attach edited JMP output in problems where you are required to use JMP or when you use the JMP output to
provide answers as part of the analysis.
Due Friday, January 29, 2016 (turn-in at the beginning of class)
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