Question

A variety of packaging solutions exist for products that must be kept within a specific temperature range. Cold chain distribution is particularly useful in the food and pharmaceutical industries. A packaging company is trying out a new packaging material that might reduce the variation of temperatures in the box. It is believed that the temperature in the box follows a normal distribution with both packaging materials. Inspectors randomly select 16 boxes of new and old packages, and report the temperatures in degrees Celsius 24 hours after they are sealed for shipment. A portion of the data is shown in the accompanying table. Assume that the two samples are drawn independently from normally distributed populations.

New Package	Old Package
5.34	5.55
5.17	4.57
5.59	6.81
5.44	5.30
5.15	3.96
5.70	3.48
6.10	6.23
5.40	3.71
6.50	5.07
5.99	5.09
5.97	4.99
5.47	5.16
4.30	5.90
5.52	4.12
6.34	3.89
5.13	5.63

Let Old Package represent population 1 and New Package represent population 2. Select the hypotheses to test whether the new packaging material reduces the variation of temperatures in the box.

• H₀: σ₁² / σ₂² = 1, H_A: σ₁² / σ₂² ≠ 1

• H₀: σ₁² / σ₂² ≤ 1, H_A: σ₁² / σ₂² > 1

• H₀: σ₁² / σ₂² ≥ 1, H_A: σ₁² / σ₂² < 1

b. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 3 decimal places.)

Test statistic________.

c. Find the p-value.

• 0.01  p-value < 0.025

• p-value < 0.01

• p-value  0.10
• 0.05  p-value < 0.10
• 0.025  p-value < 0.05

d. Make a conclusion at the 10% significance level.

• Reject H₀; we can say that new packaging material reduces the variation of temperatures.

• Reject H₀; we cannot say that new packaging material reduces the variation of temperatures..

• Do not reject H₀; we can say that new packaging material reduces the variation of temperatures.

• Do not reject H₀; we cannot say that new packaging material reduces the variation of temperatures.

Answer 1

i am using minitab to solve the problem.

steps:-

copy the data in minitab stat basic statistics 2 variance each sample is in int own column in sample 1 select old package , in sample 2 select new package options in ratio select (Sample 1 variance)/(sample 2 variance) in confidence level type 90in hypothesized ratio type 1select the alternative hypothesis as ratio > hypothesized ratio tick use test and confidence intervals based on normal distribution okok.

*** SOLUTION ***

a).hypothesis:-

b). the test statistic = 3.142

c). p value : 0.01 < p value < 0.025

d).Reject H₀; we can say that new packaging material reduces the variation of temperatures.

[ p value < 0.025 < 0.10(alpha) ]

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