Introduction to Probability and Statistics for Engineers and Scientists

284 Chapter 7: Parameter Estimation

32.LetX 1 ,...,Xn,Xn+ 1 denote a sample from a normal population whose mean

μand varianceσ^2 are unknown. Suppose that we are interested in using the

observed values ofX 1 ,...,Xnto determine an interval, called apredictioninterval,

that we predict will contain the value ofXn+ 1 with 100(1−α) percent confidence.

LetXnandSn^2 be the sample mean and sample variance ofX 1 ,...,Xn.

(a) Determine the distribution of

Xn+ 1 −Xn

(b) Determine the distribution of

Xn+ 1 −Xn

Sn

√

1 +

1

n

(c) Give the prediction interval forXn+ 1.

(d) The interval in part (c) will contain the value ofXn+ 1 with 100(1−α) percent

confidence. Explain the meaning of this statement.

33.National Safety Council data show that the number of accidental deaths due to

drowning in the United States in the years from 1990 to 1993 were (in units of

one thousand) 5.2, 4.6, 4.3, 4.8. Use these data to give an interval that will, with

95 percent confidence, contain the number of such deaths in 1994.

34.The daily dissolved oxygen concentration for a water stream has been recorded

over 30 days. If the sample average of the 30 values is 2.5 mg/liter and the sample

standard deviation is 2.12 mg/liter, determine a value which, with 90 percent

confidence, exceeds the mean daily concentration.

35.Verify the formulas given in Table 7.1 for the 100(1−α) percent lower and upper

confidence intervals forσ^2.

36.The capacities (in ampere-hours) of 10 batteries were recorded as follows:

140, 136, 150, 144, 148, 152, 138, 141, 143, 151

(a) Estimate the population varianceσ^2.

(b) Compute a 99 percent two-sided confidence interval forσ^2.

(c) Compute a valuevthat enables us to state, with 90 percent confidence, that

σ^2 is less thanv.

37.Find a 95 percent two-sided confidence interval for the variance of the diameter

of a rivet based on the data given here.