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# Statistics problems

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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 &#963; = 2 cm/s. The experimenter decided to specify a type I error probability or significance level of &#945; = 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 &#945; = 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 &#945; = 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?

https://brainmass.com/statistics/hypothesis-testing/statistics-problems-170078

#### Solution Summary

This solution gives the step by step method for basic statistics, graphs,confidence interval , hoc plot, normal probability plots

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## Statistics Problems - Regression Analysis, Autocorrelation, Multicollinearity

1. Suppose an appliance manufacturer is doing a regression analysis, using quarterly time-series data, of the factors affecting its sales of appliances. A regression equation was estimated between appliance sales (in dollars) as the dependent variable and disposable personal income and new housing starts as the independent variables. The statistical tests of the model showed large t-values for both independent variables, along with a high r2 value. However, analysis of the residuals indicated that substantial autocorrelation was present.

a. What are some of the possible causes of this autocorrelation?

b. How does this autocorrelation affect the conclusions concerning the significance of the individual explanatory variables and the overall explanatory power of the regression model?

c. Given that a person uses the model for forecasting future appliance sales, how does this autocorrelation affect the accuracy of these forecasts?

d. What techniques might be used to remove this autocorrelation from the model?

2. Suppose the appliance manufacturer discussed in Exercise 1 also developed another model, again using time-series data, where appliance sales was the dependent variable and disposable personal income and retail sales of durable goods were the independent variables. Although the r2 statistic is high, the manufacturer also suspects that serious multicollinearity exists between the two independent variables.

a. In what ways does the presence of this multicollinearity affect the results of the regression analysis?

b. Under what conditions might the presence of multicollinearity cause problems in the use of this regression equation in designing a marketing plan for appliance sales?

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