Lecturer: Mr JC Kabala
27 September 2021
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Student number:
Operations Management Techniques 1: PBT150S
Classwork – Chapter 6: Confidence Intervals - MEMO
Question 1
Solution: a)
1. Identify the sample statistics n and x.
𝑛 = 1000
𝑥 = 823
2. Find the point estimate
𝑃̂ =
𝑥
823
=
= 0.823
𝑛 1000
̂ can be approximated by a normal distribution.
3. Verify that the sampling distribution of 𝒑
𝑛𝑝̂ ≥ 5 → 1000 ∗ 0.823 = 823
𝑛𝑞̂ ≥ 5 → 1000 ∗ 0.177 = 177
4. Find the critical value zc that corresponds to the given level of confidence c.
𝑐 = 95% = 0.95
𝑧𝑐 = ±1.96
𝐺𝑟𝑎𝑝ℎ 𝑜𝑓 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑁𝑜𝑟𝑚𝑎𝑙 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
5. Find the margin of error E.
𝐸 = 𝑍𝑐 ∗ √
𝑝̂ 𝑞̂
0.823 ∗ 0.177
= 1.96 ∗ √
= 0.0237
𝑛
1000
6. Find the left and right endpoints and form the confidence interval.
𝐿𝑒𝑓𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠: 𝑝̂ − 𝐸 = 0.823 − 0.0237 = 0.7993
𝑅𝑖𝑔ℎ𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠: 𝑝̂ + 𝐸 = 0.823 + 0.0237 = 0.8467
𝐶𝑜𝑛𝑓𝑖𝑑𝑒𝑛𝑐𝑒 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙:
0.7993 < 𝑝 < 0.8467
Comment: With 95% confidence, you can say that the population proportion of cases of lung cancer
resulted in death within 10 years is between 79.93% and 84.67%.
Solution: b) PLEASE IGNORE, THE SECTION WAS NOT COVERED
𝒄 =. . . . ..
𝒁𝒄 =. . . . . .
𝐺𝑟𝑎𝑝ℎ 𝑜𝑓 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑁𝑜𝑟𝑚𝑎𝑙 𝐷𝑖𝑠𝑡𝑟𝑖𝑏𝑢𝑡𝑖𝑜𝑛
𝒁𝒄 𝟐
̂𝒒
̂ ( ) =. . . . ..
𝒏=𝒑
𝑬
𝑬 =. . . . ..
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Lecturer: Mr JC Kabala
27 September 2021
Question 2
Solution
1. Verify that the population has a normal distribution.
The lifetimes of batteries are normally distributed as indicated in the scenario.
2. Identify the sample statistic n and the degrees of freedom.
𝑛=5
𝑑𝑓 = 𝑛 − 1 = 5 − 1 = 4
3. Find the point estimate s2.
𝑆2 =
∑(𝑥𝑖 − 𝑥̅ )2 1.2100 + 0.3600 + 0.0000 + 0.2500 + 1.4400 3.26
=
=
= 0.8150
𝑛−1
5−1
4
4. Find the critical values χ2R and χ2L that correspond to the given level of confidence c.
Chi square distribution
𝜒𝐿2 = 0.484
𝜒𝑅2 = 11.143
5. Find the left and right endpoints and form the confidence interval for the population variance.
𝐿𝑒𝑓𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠:
(𝑛 − 1)𝑆 2 (5 − 1) ∗ 0.815
=
= 0.2926
11.143
𝜒𝑅2
(𝑛 − 1)𝑆 2 (5 − 1) ∗ 0.815
𝑅𝑖𝑔ℎ𝑡 𝑒𝑛𝑑𝑝𝑜𝑖𝑛𝑡𝑠:
=
= 6.7355
0.484
𝜒𝑅2
𝑪𝒐𝒏𝒇𝒊𝒅𝒆𝒏𝒄𝒆 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍: 0.2926 < 𝝈𝟐 < 6.7355
Comment: With 95% confidence, you can conclude that the population variance of the battery’s lifetime is
between 0.2926 and 6.7355 years.
Manufacturer’s claim: The manufacturer’s claim that 𝜎 2 = 1 is valid.
Question 3
3.1.
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Lecturer: Mr JC Kabala
27 September 2021
3.2.
Question 4
4.1.
4.2.
Question 5
5.1. Of 1000 randomly selected cases of lung cancer, 823 resulted in death within 10 years. Construct a 95%
two-sided confidence interval on the death rate from lung cancer. How large a sample would be required
to be at least 95% confident that the error in estimating the 10-year death rate from lung cancer is within
0.03? See question 1.
5.2. The Arizona Department of Transportation wishes to survey state residents to determine what proportion
of the population would like to increase state wide highway speed limits to 75 mph from 65 mph. How
many residents do they need to survey if they want to be at least 99% confident that the sample proportion
is within 0.05 of the true proportion? Please ignore, the section was not covered.
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Lecturer: Mr JC Kabala
27 September 2021
Question 6
6.1.
6.2.
Question 7
7.1.
7.2.
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Lecturer: Mr JC Kabala
27 September 2021
Question 8
A manufacturer of car batteries claims that the batteries will last, on average, 3 years with a variance of 1 year.
If 5 of these batteries have lifetimes of 1.9, 2.4, 3.0, 3.5, and 4.2 years, construct a 95% confidence interval
for 𝜎 2 and decide if the manufacturer’s claim that 𝜎 2 = 1 is valid. Assume the population of battery lives to
be approximately normally distributed. See question 2.
Question 9
A machine produces metal pieces that are cylindrical in shape. A sample of pieces is taken, and the diameters
are found to be 1.01, 0.97, 1.03, 1.04, 0.99, 0.98, 0.99, 1.01, and 1.03 centimeters. Find a 99% confidence
interval for the mean and variance diameter of pieces from this machine, assuming an approximately normal
distribution. PLEASE COMPLETE
Question 10
A random sample of 10 chocolate energy bars of a certain brand has, on average, 230 calories per bar, with a
standard deviation of 15 calories. Construct a 99% confidence interval for the true mean and standard deviation
calorie content of this brand of energy bar. Assume that the distribution of the calorie content is approximately
normal. PLEASE COMPLETE
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