Download and , find a 90% confidence interval for

Estimating Confidence Intervals (7.4) Worksheet
1. Consider two independent normal distributions. A random sample of size 𝑛1 = 20
from the first distribution showed 𝑥̅1 = 12 and a random sample size of 𝑛2 = 25
from the second distribution showed 𝑥̅2 = 14.
a) Given 𝜎1 = 3 and 𝜎2 = 4, find a 90% confidence interval for 𝜇1 − 𝜇2
b) Given 𝜎1 and 𝜎2 are unknown, 𝑠1 = 3 and 𝑠2 = 4 find a 90% confidence interval
for 𝜇1 − 𝜇2
2. Consider two independent binomial experiments. In the first one, 40 trials had 10
successes. In the second one, 60 trials had 6 successes.
a) Is it appropriate to use a normal distribution to approximate the 𝑝̂1 − 𝑝̂ 2
distribution ?
b) Find a 90% confidence interval for 𝑝̂1 − 𝑝̂2 .
3. Inorganic phosphorous is a naturally occurring element in all plants and animals.
Archaeologists look at geochemical surveys of soil samples to look at phosphorous
content to determine if an ancient burial site, food storage, or village could have
existed at that site. Two independent random sample are taken. Assume the
population distributions of phosphorous are mound – shaped and symmetric for the
two samples.
Sample 1 : 𝑥1 ; 𝑛1 = 12
540 810 790 790 340 800
890 860 820 640 970 720
Sample 2 : 𝑥2 ; 𝑛2 = 16
750 870 700 810 965 350 895 850
635 955 710 890 520 650 280 993
a) Using a calculator, find the mean and standard deviation for both samples
𝑥̅1 = ___________
𝑥̅2 = __________
𝑠1 = __________
𝑠2 = __________
b) Let 𝜇1 be the population mean for 𝑥1 and 𝜇2 be the population mean for 𝑥2 .
Find a 90% confidence interval for 𝜇1 − 𝜇2 .
4. Independent random samples of professional football and basketball players gave
the following information.
Football Players
Basketball Players
𝑥̅1 ≈ 259.6 lbs
𝑥̅2 ≈ 205.8 lbs
𝑠1 ≈ 12.1 lbs
𝑠2 ≈ 12.9 lbs
𝑛1 = 21
𝑛2 = 19
Let 𝜇1 be the population mean for 𝑥1 and 𝜇2 be the population mean for 𝑥2 .
Find a 90% confidence interval for 𝜇1 − 𝜇2 .
5. Isabel Myers was a pioneer in the study of personality types. She identified four
personality preferences. Marriage counselors know that couples who have none of
the four preferences in common may have stormy marriages. In one random sample
of 375 married couples, Myers found that 289 had two or more personality
preferences in common. In another random sample of 571 married couples, Myers
found that only 23 had no preferences in common. Let 𝑝1 be the population
proportion of all married couples who have two or more personality preferences in
common. Let 𝑝2 be the population proportion of all married couples who have no
personality preferences in common.
a) Can a normal distribution be used to approximate the 𝑝̂1 − 𝑝̂ 2 distribution ?
b) Find a 99% confidence interval for 𝑝1 − 𝑝2 .
6. At community Hospital, the burn center is experimenting with a new plasma
compress treatment. A random sample 𝑛1 = 316 patients with minor burns
received the new plasma compress treatment. Of these patients, it was found that
259 had no visible scars after treatment. Another random sample of 𝑛2 = 419
patients with minor burns received no plasma compress treatment. For this group,
it was found that 94 had no visible scars after treatment. Let 𝑝1 be the population
proportion of all patients with minor burns receiving the plasma treatment who
have no visible scars. Let 𝑝2 be the population proportion of all patients with minor
burns not receiving the plasma treatment who have no visible scars.
a) Can a normal distribution be used to approximate the 𝑝̂1 − 𝑝̂2 distribution ?
b) Find a 95% confidence interval for 𝑝1 − 𝑝2 .