# Normal probability, Confidence interval & Sample size

These are some of the questions I've run into and need help for. I'm unable to solve some of these problems. I'm studying for my final next week and I'm preparing for that.

7.16 Daily output of Marathon's Garyville, Louisiana, refinery is normally distributed with a mean of

232,000 barrels of crude oil per day with a standard deviation of 7,000 barrels. (a) What is the

probability of producing at least 232,000 barrels? (b) Between 232,000 and 239,000 barrels?

(c) Less than 239,000 barrels? (d) Less than 245,000 barrels? (e) More than 225,000 barrels?

7.17 Assume that the number of calories in a McDonald's Egg McMuffin is a normally distributed

random variable with a mean of 290 calories and a standard deviation of 14 calories. (a) What is

the probability that a particular serving contains fewer than 300 calories? (b) More than 250 calories?

(c) Between 275 and 310 calories? Show all work clearly. (Data are from McDonalds.com)

7.18 The weight of a miniature Tootsie Roll is normally distributed with a mean of 3.30 grams and

standard deviation of 0.13 grams. (a) Within what weight range will the middle 95 percent of all

miniature Tootsie Rolls fall? (b) What is the probability that a randomly chosen miniature

Tootsie Roll will weigh more than 3.50 grams? (Data are from a project by MBA student Henry

Scussel.)

7.19 The time required to verify and fill a common prescription at a neighborhood pharmacy is normally

distributed with a mean of 10 minutes and a standard deviation of 3 minutes. Find the time

for each event. Show your work.

a. Highest 10 percent b. Highest 50 percent c. Highest 5 percent

d. Highest 80 percent e. Lowest 10 percent f. Middle 50 percent

g. Lowest 93 percent h. Middle 95 percent i. Lowest 7 percent

7.21 The weight of newborn babies in Foxboro Hospital is normally distributed with a mean of

6.9 pounds and a standard deviation of 1.2 pounds. (a) How unusual is a baby weighing

8.0 pounds or more? (b) What would be the 90th percentile for birth weight? (c) Within what

range would the middle 95 percent of birth weights lie?

8.6 Prof. Hardtack gave three exams last semester in a large lecture class. The standard deviation

Ï? = 7 was the same on all three exams, and scores were normally distributed. Below are scores

for 10 randomly chosen students on each exam. Find the 95 percent confidence interval for

the mean score on each exam. Do the confidence intervals overlap? If so, what does this suggest?

8.7 In a certain manufacturing process, the diameter of holes drilled in a steel plate is a normally

distributed random variable. The process standard deviation is known to be Ï?=0.005 cm.A sample

of 15 plates shows a mean hole diameter of 2.475 cm. Find the 95 percent confidence interval for ?.

8.11 A sample of 21 minivan electrical warranty repairs for loose, not attached wires (one of several

electrical failure categories the dealership mechanic can select) showed a mean repair cost of

$45.66 with a standard deviation of $27.79. (a) Construct a 95 percent confidence interval for the

true mean repair cost. (b) How could the confidence interval be made narrower? (Data are from a

project by MBA student Tim Polulak.)

8.13 A random sample of monthly rent paid by 12 college seniors living off campus gave the results

below (in dollars). Find a 99 percent confidence interval for ?, assuming that the sample is from

a normal population. Rents

900 810 770 860 850 790

810 800 890 720 910 640

8.19 From a list of stock mutual funds, 52 funds were selected at random. Of the funds chosen, it was

found that 19 required a minimum initial investment under $1,000. (a) Construct a 90 percent confidence

interval for the true proportion requiring an initial investment under $1,000. (b) May normality

be assumed? Explain.

8.22 A sample of 50 homes in a subdivision revealed that 24 were ranch style (as opposed to colonial,

tri-level, or Cape Cod). (a) Construct a 98 percent confidence interval for the true proportion of

ranch style homes. (b) Check the normality assumption.

8.27 Popcorn kernels are believed to take between 100 and 200 seconds to pop in a certain microwave.

What sample size (number of kernels) would be needed to estimate the true mean seconds to pop

with an error of ± 5 seconds and 95 percent confidence? Explain your assumption about Ï?.

8.28 Analysis showed that the mean arrival rate for vehicles at a certain Shell station on Friday afternoon

last year was 4.5 vehicles per minute. How large a sample would be needed to estimate this

yearâ??s mean arrival rate with 98 percent confidence and an error of ±0.5?

8.31 (a) What sample size would be needed to estimate the true proportion of American households

that own more than one DVD player, with 90 percent confidence and an error of ±0.02? (b) What

sampling method would you recommend? Why?

8.33 (a) What sample size would be needed to estimate the true proportion of American adults who

know their cholesterol level, using 95 percent confidence and an error of ±0.02? (b) What sampling

method would you recommend, and why?

https://brainmass.com/math/probability/normal-probability-confidence-interval-sample-size-329862

#### Solution Summary

The solution provides step by step method for the calculation of Normal probability, confidence interval and sample size. Formula for the calculation and Interpretations of the results are also included. Interactive excel sheet is also included.

Statistics: Sample variability, and Introduction to statistical inferences

1. Consider a normal population with µ = 25 and σ = 8.0.

(A) Calculate the standard score for a value x of 27.

(B) Calculate the standard score for a randomly selected sample of 30 with = 27.

(C) Explain why the standard scores of 27 are different between A and B above

2. Assume that the mean score on a certain aptitude test across the nation is 100, and that the standard deviation is 20 points. Find the probability that the mean aptitude test score for a randomly selected group of 150 8th graders is between 99.5 and 100.5.

3. Assume that a sample is drawn and z(α/2) = 1.96 and σ = 20. Answer the following questions:

(A) If the Maximum Error of Estimate is 0.02 for this sample, what would be the sample size?

(B) Given that the sample Size is 400 with this same z(α/2) and σ, what would be the Maximum Error of Estimate?

(C) What happens to the Maximum Error of Estimate as the sample size gets smaller?

(D) What effect does the answer to C above have to the size of the confidence interval?

4. By measuring the amount of time it takes a component of a product to move from one workstation to the next, an engineer has estimated that the standard deviation is 4.17 seconds.

Answer each of the following (show all work):

(A) How many measurements should be made in order to be 95% certain that the maximum error of estimation will not exceed 0.5 seconds?

(B) What sample size is required for a maximum error of 2.0 seconds?

5. A 98% confidence interval estimate for a population mean was computed to be (36.5, 52.9). Determine the mean of the sample, which was used to determine the interval estimate (show all work).

6. A study was conducted to estimate the mean amount spent on birthday gifts for a typical family having two children. A sample of 160 was taken, and the mean amount spent was $223.24. Assuming a standard deviation equal to $49.78, find the 95% confidence interval for , the mean for all such families (show all work).

7. A confidence interval estimate for the population mean is given to be (39.86, 47.87). If the standard deviation is 16.219 and the sample size is 63, answer each of the following (show all work):

(A) Determine the maximum error of the estimate, E.

(B) Determine the confidence level used for the given confidence interval