Sampling Distribution and Point Estimation of Parameters, Statistical Intervals

The television picture tubes of manufacturer A have a mean lifetime of 6.5 years and a standard deviation of 0.9 year, while those of manufacturer B have a mean lifetime of 6.0 years and a standard deviation of 0.8 year. What is the probability that a random sample of 36 tubes from manufacturer A will have a mean lifetime that is at least 1 year more than the mean lifetime of a sample of 49 tubes from manufacturer B? Given the following information.

0. 0040

An electrical firm manufactures light bulbs that have a length of life that is approximately normally distributed, with mean equal to 800 hours and a standard deviation of 40 hours. Find the probability that a random sample of 16 bulbs will have an average life of less than 775 hours.

0. 0062

The length of the battery of the 2D echo machine is approximately normally distributed, with mean equal to 500 hours and standard deviation of 20 hours. Find the probability that a random sample of 10 batteries will have an average of less than 483 hours.

0. 0036

When a production machine is properly calibrated, it requires an average of 25 seconds per unit produced, with a standard deviation of 3 seconds. For a simple random sample of n = 36 units, the sample mean is found to be 26.2 seconds per unit. When the machine is properly calibrated, what is the probability that the mean for a simple random sample of this size will be at least 26.2 seconds?

0. 0082

The campaign manager for a political candidate claims that 55% of registered voters favor the candidate over her strongest opponent. Assuming that this claim is true, what is the probability that in a simple random sample of 300 voters, at least 60% would favor the candidate over her strongest opponent?

0. 0409

12% of students at NCSU are left-handed. What is the probability that in a sample of 100 students, the sample proportion that are left-handed is less than 11%?

0. 3873

In 2000, the GSS asked: Are you willing to pay much higher prices in order to protect the environment? Of n = 1154 respondents, 518 were willing. Find the 95% CI for the population that is willing.

0. 4203 ≤ p ≤ 0. 4775

The mean height of 15-year-old boys is 175 cm and the variance is 64. For girls, the mean is 165 and the variance is 64. If 8 boys and 8 girls were sampled, what is the probability that the mean height of the sample of boys would be at least 6 cm higher than the mean height of the sample of girls?

0. 8413

The mean time to complete a task is 727 millisecond for 3rd graders and 532 milliseconds for 5th graders. The variances of the two grades are 12,000 for 3rd graders and 10,000 for 5th graders. The times for both grades are normally distributed. You randomly sample 12 3rd graders and 14 5th graders. What is the probability that the mean time of the 3rd graders will exceed the mean time of the 5th graders by 150 msec or more?

0. 8621

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 diameter of pieces from this machine, assuming an approximately normal distribution.

0. 98 ≤ μ ≤ 1. 04

A random sample of size 25 is taken from a normal population having a mean of 80 and a standard deviation of 5. A second random sample of size 36 is taken from a different normal population having a mean of 75 and a standard deviation of 3. Find the probability that the sample mean computed from the 25 measurements will exceed the sample mean computed from the 36 measurements by at least 3.4 but less than 5.9.

0.7117

A random sample of 20 nominally measured 2mm diameter steel ball bearings is taken and the diameters are measured precisely. The measurements, in mm, are as follows: 2.02 1.94 2.09 1.95 1.98 2.00 2.03 2.04 2.08 2.07 1.99 1.96 1.99 1.95 1.99 1.99 2.03 2.05 2.01 2.03 Assuming that the diameters are normally distributed with unknown mean, μ, and unknown variance σ2, (a) find a two-sided 95% confidence interval for the variance.

1. 10x10^−3 ≤ σ2 ≤ 4. 05x10^−3

An article in the Journal of Materials Engineering describes the results of tensile adhesion tests on 22 U-700 alloy specimens. Find the 95% confidence interval (CI). The load at specimen failure is as follows (in mega pascal): 19.8, 10.1, 14.9, 7.5, 15.4, 15.4, 15.4, 18.5, 7.9, 12.7, 11.9, 11.4, 11.4, 14.1, 17.6, 16.7, 15.8, 19.5, 8.8, 13.6, 11.9, 11.4

12. 14 ≤ μ ≤ 15. 28

Consider the CVN test described in the previous example and suppose that we want to determine how many specimens must be tested to ensure that the 95% CI on μ for A238 steel cut at 60°C has a length of at most 1.0 J. Assume the bound on error in estimation E is one-half of the length of the CI.

16

Due to the decrease in interest rates, the First Citizens Bank received a lot of mortgage applications. A recent sample of 50 mortgage loans resulted in an average loan amount of $257,300. Assume a population standard deviation of $25,000. For the next customer who fills out a mortgage application, find a 95% prediction interval for the loan amount.

207,812.43 ≤ Xo ≤ 306,787.57

Suppose the following data values below are selected randomly from a sample population of normally distributed values. Find the interval estimate using 90% CL. 40 39 51 42 43 48 48 45 44 39 57 43 54

42. 81 ≤ μ ≤ 48. 43

ASTM Standard E23 defines standard test methods for notched bar impact testing of metallic materials. The Charpy V-notch (CVN) technique measures impact energy and is often used to determine whether or not a material experiences a ductile-to brittle transition with decreasing temperature. Ten measurements of impact energy (J) on specimens of A238 steel cut at 60∘C are as follows: 64.1, 64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, and 64.3. Assume that impact energy is normally distributed with σ = 1 J. Find a 95% CI for μ, the mean impact energy?

63. 84 ≤ μ ≤ 65. 08

The same data for impact testing are used to construct a lower, one-sided 95% confidence interval for the mean impact energy. *Ten measurements of impact energy (J) on specimens of A238 steel cut at 60∘C are as follows: 64.1, 64.7, 64.5, 64.6, 64.5, 64.3, 64.6, 64.8, 64.2, and 64.3 with σ = 1J. What is the interval?

63. 94 ≤ μ

A meat inspector has randomly selected 30 packs of 95% lean beef. The sample resulted in a mean of 96.2% with a sample standard deviation of 0.8%. Find a 99% prediction interval for the leanness of a new pack. Assume normality. α/2 = 0.01/2 = 0.005; df =29

93.96% < Xo ≤ 98.44%

A meat inspector has randomly selected 30 packs of 95% lean beef. The sample resulted in a mean of 96.2% with a sample standard deviation of 0.8%. Find a tolerance interval that gives two-sided 95% bounds on 90% of the distribution of packages of 95% lean beef. Assume the data came from an approximately normal distribution.

94.49 and 97.91

The average zinc concentration recovered from a sample of measurements taken in 36 different locations in a river is found to be 2.6 grams per milliliter. Find the 95% and 99% confidence intervals for the mean zinc concentration in the river. Assume that the population standard deviation is 0.3 gram per milliliter.

95% -- 2. 50 ≤ μ ≤ 2. 70 99% -- 2. 47 ≤ μ ≤ 2. 73

The amount of time that a drive-through bank teller spends on a customer is a random variable with a mean 3.2 minutes and a standard deviation 1.6 minutes. If a random sample of 64 customers is observed, find the probability that their mean time at the teller's counter is (a) at most 2.7 minutes; (b) more than 3.5 minutes; (c) at least 3.2 minutes but less than 3.4 minutes

a. 0. 0062 b. 0. 0668 c. 0. 3413

In a typical car, bell housings are bolted to crankcase castings by means of a series of 13 mm bolts. A random sample of 12 bolt-hole diameters is checked as part of a quality control process and found to have a variance of 0.0013 mm2. (a) Construct the 95% confidence interval for the variance of the holes. (b) Find the 95% confidence interval for the standard deviation of the holes

a. 6. 52x10^−4 ≤ σ2 ≤ 3. 75x10^−3 b. 0. 0255 ≤ σ ≤ 0. 0612

Let X be the height of a randomly chosen individual from a population. In order to estimate the mean and variance of X, we observe a random sample X1, X2,⋯⋯, X7. We obtain the following values (in centimeters): 166.8 171.4 169.1 178.5 168.0 157.9 170.1 Find the values of the sample mean, the sample variance, and the sample standard deviation for the observed sample.

x̄ = 168. 83 s2 = 37.68 s = 6. 14

In a psychological testing experiment, 25 subjects are selected randomly and their reaction time, in seconds, to a particular stimulus is measured. Past experience suggests that the variance in reaction times to these types of stimuli is 4 sec2 and that the distribution of reaction times is approximately normal. The average time for the subjects is 6.2 seconds. Give an upper 95% bound for the mean reaction time.

μ ≤ 6. 86

An automatic filling machine is used to fill bottles with liquid detergent. A random sample of 20 bottles results in a sample variance of fill volume of s2 = 0.01532 (fluid ounce). If the variance of fill volume is too large, an unacceptable proportion of bottles will be under- or overfilled. We will assume that the fill volume is approximately normally distributed. Find the confidence bound at 95%

σ ≤ 0. 17 fluid ounce