Page 260 - Applied statistics and probability for engineers

238 Chapter 6/Descriptive Statistics

Mind-Expanding Exercises

6-129. Consider the airfoil data in Exercise 6-18. Subtract 6-136. Suppose that you have a sample x x 2 ,… , x n and
1 ,
30 from each value and then multiply the resulting quantities have calculated x n and s n 2 for the sample. Now an (n + 1)
2
by 10. Now compute s for the new data. How is this quan- st observation becomes available. Let x n + 1 and s 2 be the
n + 1
2
tity related to s for the original data? Explain why. sample mean and sample variance for the sample using all
n 2
=
6-130. Consider the quantity ∑ i 1 (x i − ) a . For what n + 1 observations.
value of a is this quantity minimized? (a) Show how x n + 1 can be computed using x n and x n + 1 .
(
6-131. Using the results of Exercise 6-130, which of the n x n+ − ) 2
n 2 and ∑ n − ) μ 2 (b) Show that n n s + = ( 2 1 x n
2
= (x i i = 1 (x i n +1
two quantities ∑ i 1 − ) x will be 1 n − ) 1 n s +
smaller, provided that x ≠ μ? (c) Use the results of parts (a) and (b) to calculate the new
6-132. Coding the Data. Let y i = a + bx , i i = 1 2,… n , , sample average and standard deviation for the data of
,
where a and b are nonzero constants. Find the relationship Exercise 6-38, when the new observation is x 38 = 64.
between x and y, and between s x and s y .
6-133. A sample of temperature measurements in a fur- 6-137. Trimmed Mean. Suppose that the data are arranged
nace yielded a sample average (°F) of 835.00 and a sample in increasing order, T% of the observations are removed from
standard deviation of 10.5. Using the results from Exercise each end, and the sample mean of the remaining numbers is
6-132, what are the sample average and sample standard calculated. The resulting quantity is called a trimmed mean,
o
deviations expressed in C? which generally lies between the sample mean x and the sam-
6-134. Consider the sample x x 2 ,… , x n with sam- ple median x. Why? The trimmed mean with a moderate trim-
1 ,
ple mean x and sample standard deviation s . Let ming percentage (5% to 20%) is a reasonably good estimate of
x − ) / s i = , ,… the middle or center. It is not as sensitive to outliers as the mean
z i = ( i x , 1 2 , n. What are the values of the
but is more sensitive than the median.
sample mean and sample standard deviation of the z i?
(a) Calculate the 10% trimmed mean for the yield data in
6-135. An experiment to investigate the survival time in
Exercise 6-33.
hours of an electronic component consists of placing the
(b) Calculate the 20% trimmed mean for the yield data in
parts in a test cell and running them for 100 hours under
Exercise 6-33 and compare it with the quantity found
elevated temperature conditions. (This is called an “accel-
in part (a).
erated” life test.) Eight components were tested with the
(c) Compare the values calculated in parts (a) and (b) with the
following resulting failure times:
sample mean and median for the yield data. Is there much
+
,
75 63 100 , 36 51 45 80 90 difference in these quantities? Why?
,
,
,
,
,
+
The observation 100 indicates that the unit still functioned at 6-138. Trimmed Mean. Suppose that the sample size n is
100 hours. Is there any meaningful measure of location that such that the quantity nT /100 is not an integer. Develop a
can be calculated for these data? What is its numerical value? procedure for obtaining a trimmed mean in this case.
Important Terms and Concepts
Box plot Multivariate data Probability plot Sample standard deviation
Degrees of freedom Normal probability plot Relative frequency Sample variance
Frequency distribution and Outlier distribution Scatter diagram
histogram Pareto chart Sample correlation coeficient Stem-and-leaf diagram
Histogram Percentile Sample mean Time series
Interquartile range Population mean Sample median
Matrix of scatter plots Population standard deviation Sample mode
Quartiles, and percentiles Population variance Sample range