In: Statistics and Probability
A pharmaceutical manufacturer purchases a particular material from two different suppliers. The mean level of impurities in the raw material is approximately the same for both suppliers, but the manufacturer is concerned about the variability of the impurities from shipment to shipment. To compare the variation in percentage impurities for the two suppliers, the manufacturer selects 9 shipments from each of the two suppliers and measures the percentage of impurities in the raw material for each shipment. The sample means and variances are shown in the table.
Supplier A | Supplier B |
x1 = 1.87 |
x2 = 1.83 |
s12 = 0.271 |
s22 = 0.092 |
n1 = 9 |
n2 = 9 |
(a) Do the data provide sufficient evidence to indicate a
difference in the variability of the shipment impurity levels for
the two suppliers? Test using α = 0.01. Based on the
results of your test, what recommendation would you make to the
pharmaceutical manufacturer?
State the null and alternative hypotheses.
H0: σ12 = σ22 versus Ha: σ12 < σ22
H0: σ12 = σ22 versus Ha: σ12 ≠ σ22
H0: σ12 < σ22 versus Ha: σ12 > σ22
H0: σ12 ≠ σ22 versus Ha: σ12 = σ22
H0: σ12 = σ22 versus Ha: σ12 > σ22
State the test statistic. (Round your answer to two decimal
places.)
F =
State the rejection region. (Round your answer to two decimal
places.)
F >
State the conclusion.
H0 is rejected. There is insufficient evidence to indicate that the supplier's shipments differ in variability.
H0 is rejected. There is sufficient evidence to indicate that the supplier's shipments differ in variability.
H0 is not rejected. There is sufficient evidence to indicate that the supplier's shipments differ in variability.
H0 is not rejected. There is insufficient evidence to indicate that the supplier's shipments differ in variability.
(b) Find a 99% confidence interval for
σ22. (Round your answers to three
decimal places.)
Interpret your results.
In repeated sampling, 1% of all intervals constructed in this manner will enclose σ22.
There is a 1% chance that an individual sample variation will fall within the interval limits.
There is a 99% chance that an individual sample variation will fall within the interval.99% of all values will fall within the interval limits.
In repeated sampling, 99% of all intervals constructed in this manner will enclose σ22.
The statistical software output for this problem is:
Hence,
a) Hypotheses: H0: σ12 = σ22 versus Ha: σ12 ≠ σ22
Test statistic = 2.95
Rejection region: F > 6.03
Conclusion: H0 is not rejected. There is insufficient evidence to indicate that the supplier's shipments differ in variability.
b) 99% confidence interval:
(0.034, 0.548)
Interpretation: In repeated sampling, 99% of all intervals constructed in this manner will enclose σ22.