Question
9. Chebyshev's rule states that 68% of the observations on a variable will lie within plus or minus two standard deviations from the mean value
9. Chebyshev's rule states that 68% of the observations on a variable will
lie within plus or minus two standard deviations from the mean value for
that variable. True or False. Explain your answer fully.
10. A manufacturer of automobile batteries claims that the average length
of life for its grade A battery is 60 months. But the guarantee on this brand
is for just 36 months. Suppose that the frequency distribution of the lifelength data is unimodal and symmetrical and that the standard deviation is
known to be 10 months. Suppose further that your battery lasts 37 months.
What could you infer, if anything, about the manufacturer's claim?
11. At one university, the students are given z-scores at the end of each
semester rather than the traditional GPA's. The mean and standard deviations of all students' cumulative GPA's on which the z-scores are based
are 2.7 and 0.5 respectively. Students with z-scores below -1.6 are put on
probation. What is the corresponding probationary level of the GPA?
12. Two variables have identical standard deviations and a covariance equal
to half that common standard deviation. If the standard deviation of the
two variables is 2, what is the correlation coefficient between them?
13. Application of Chebyshev's rule to a data set that is roughly symmetrically distributed implies that at least one-half of all the observations lie in
the interval from 3.6 to 8.8. What are the approximate values of the mean
and standard deviation of this data set?
14. The number of defective items in 15 recent production lots of 100 items
each were as follows:
3, 1, 0, 2, 24, 4, 1, 0, 5, 8, 6, 3, 10, 4, 2
a) Calculate the mean number of defectives per lot. (4.87)
b) Array the observations in ascending order. Obtain the median of this
data set. Why does the median differ substantially from the mean
here? Obtain the range and the interquartile range. (3, 24, 4)
c) Calculate the variance and the standard deviation of the data set.
Which observation makes the largest contribution to the magnitude of
the variance through the sum of squared deviations? Which observation makes the smallest contribution? What general conclusions are
implied by these findings?
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