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Probability, Test Statistics and Error

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In a recent year, scores on a standardized test for high schools student with a 3.50 to 4.00 grade point average were normally distributed, with a mean of 37.9 and a standard deviation of 2.4. A student with a 3.50 to 4.00 grade point average who took the standardized test is randomly selected.

Four Decimal Places for Questions

Find the probability of a student test score less than 36
The probability of a student scoring less than 36 is ________

Find the probability of a student test score between 35.1 and 40.7
The probability of a student scoring between 35.1 and 40.7 is ________

Find the probability of a student test score more than 39.3
The probability of a student scoring more than 39.3 is ___________

_________________________________

*A microwave oven repairer says that the mean repair cost for damaged microwave ovens is less than $110. You work for the repairer and want to test this claim. You find that a random sample of five microwave ovens has a mean repair cost of $115 and a standard deviation of $12.50. At a=0.05, do you have enough to support the repairer's claim? Assume the population is normally distributed.

What is (are) the critical value(s), t0? t0 = _________

Find the standardized test statistic t = __________

*Use the confidence interval to find the estimated margin of error. Then find the sample mean. A biologist reports a confidence interval of (2,9,4,5) when estimating the mean height (in centimeters) of a sample seedlings.

A. The estimated margin of error is _______
B. The Sample mean is ________

_____________________________________________
*Construct the 95% and 99% confidence intervals for the population proportion "p" using the sample statistics below. Which interval is wider? If convenient, use technology to construct the confidence intervals.

Three decimal places as needed
95% confidence intervals for the population proportion "p" is _____ - ______
99% confidence intervals for the population proportion "p" is _____ - ______
Which interval is wider? 95% confidence intervals or 99% confidence intervals

__________________________________________

*Find the mean, variance, and standard deviation of the binomial distribution with the given values of n and p: n=80, p=0.5

The mean, u, is __________ (nearest tenth)
Variance, 2 is_______ (nearest tenth)
The standard deviation, , is ________ (nearest tenth)
______________________________________________

*A frequency distribution is shown below. Complete part (a) through (e)

The number of dogs per household in a small town
Dog 0 1 2 3 4 5
Household 1370 406 163 50 26 16

0 ___ 1___2___3___4___5___ (nearest thousand)
Find the mean of the probability distribution: u=______ (nearest tenth)
Find the variance of the probability distribution: 2 = ________(nearest tenth)
Find the standard deviation of the probability distribution: = ___(nearest tenth)

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Solution Summary

The following helps calculate problems involving critical value, substantiatzed

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The probability of a successful firing is, on any test, 0.95.

1. Five cruise missiles have been built by an aerospace company. The probability of a successful firing is, on any test, 0.95. Assuming independent firings, what is the probability that the first failure occurs on the fifth firing?

2. A lot of 25 color television tubes is subjected to an acceptance testing procedure. The procedure consists of drawing five tubes at random, without replacement, and testing them. If two or fewer tubes fail, the remaining ones are accepted. Otherwise, the lot is rejected. Assume the lot contains four defective tubes. What is the exact probability of lot acceptance?

3. Aircrew escape systems are powered by a solid propellant. The burning rate of this propellant is an important product characteristic. Specifications require that the mean burning rate must be 50 cm/s. We know that the standard deviation of burning rate is σ = 2 cm/s. The experimenter decided to specify a type I error probability or significance level of α = 0.05. He selects a random sample of n = 25 and obtains a sample average burning rate of = 51.3 cm/s.

a. Test the hypothesis that the mean burning rate is 50 cm/s.
b. What is the P-value for this test?
c. Construct a 95% two-sided confidence interval on mean burning rate.

4. 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 = 0.0153 square fluid ounces. If the variance of fill volume exceeds 0.01 square fluid ounces, an unacceptable proportion of bottles will be under and overfilled. Is there evidence in the sample data to suggest that the manufacturer has a problem with under- and overfilled bottles? Use α = 0.05, and assume that fill volume has a normal distribution.
5. An experiment was performed to determine the effect of four different chemicals on the strength of a fabric. These chemicals are used as part of the permanent press finishing process. Five fabric samples were selected, and a randomized complete block design was run by testing each chemical type once in random order on each fabric sample. The data are shown below.

Chemical Type Fabric Strength
1 2 3 4 5
1 1.3 1.6 0.5 1.2 1.1
2 2.2 2.4 0.4 2.0 1.8
3 1.8 1.7 0.6 1.5 1.3
4 3.9 4.4 2.0 4.1 3.4

a. Perform an analysis of variance to test whether the differences among the sample means of the chemical types are significant. Use α = 0.05
b. Construct box plots of fabric strengths by chemical types.
c. Construct a normal probability plot of the residuals from this experiment. Does the assumption of a normal distribution for fabric strength seem reasonable?

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