Part One: Whirlpool Calypso™ • Name:

Three years ago the Whirlpool Corporation was designing their new Whirlpool Calypso™ washing machine. Without an agitator that twists back and forth in the center of the wash basket, the Calypso washer cleans clothes an entirely new way. A "Nutate" wash motion gently bounces clothes through a continuous shower of detergent and water. The action is kinder to fabrics because it's designed to provide better cleaning with less stress to garment fibers. The new washing machine uses 50% less water and 66% less energy than the average conventional washer and delivers the largest usable capacity of any high efficiency vertical axis washer. The Whirlpool Corporation contracted MSC.Adams Software to design a new brake to stop the wash basket. Developing the brake was a challenge on this washer because it spins at 800 rpm, as opposed to 640 rpm for Whirlpool Corporation's traditional washers, and it also has a larger capacity that encourages consumers to load it with bigger wash loads. The brakes on early prototype washers didn't meet the company's tough product specifications. As a result of Computer Aided Design, a new brake was designed and tested in a prototype washer. The time in seconds to stop the basket for twenty trial runs of the prototype were measured.

For the brake time:

1. _________ Determine the sample size n.
2. _________ Calculate the sample mean x.
3. _________ Determine the median.
4. _________ Determine the mode.
5. _________ Determine the minimum.
6. _________ Determine the maximum.
7. _________ Calculate the range.
8. _________ Calculate the sample standard deviation sx.
9. _________ Calculate the sample Coefficient of Variation.
10. _________ Determine the class width. Use 5 bins (classes or intervals)
11. Fill in the following table with the class upper limits in the first column, the frequencies in the second column, and the relative frequencies in the third column

    Bins	Frequency	Relative Frequency f/n
    _________	_________	_________
    _________	_________	_________
    _________	_________	_________
    _________	_________	_________
    _________	_________	_________
    Sums:	_________	_________

12. Sketch a histogram of the relative frequency data on the back of the paper.
13. _________ What is the shape of the distribution?
14. _________ Using the relative frequencies in the table above, what is the probability that a brake will require 5.82 seconds or more to stop?
15. Construct a 95% confidence interval for the population mean µ brake time. Note that n is less than 30. Use the sample mean and sample standard deviation to generate your error tolerance E.
    1. __________ How many degrees of freedom?
    2. __________ What is t_c?
    3. The error tolerance E = _______________
    4. The 95% confidence interval for µ is ____________ ≤ µ ≤ ____________
16. __________ The early prototype brakes that did not meet specifications took 5.75 seconds to bring the basket to a stop. Based on the above data and a 95% confidence level, are the new brakes statistically significantly faster at stopping the wash basket?
17. __________ Using 5.75 seconds as the population mean and the sample size, sample mean and sample standard deviation from questions one, two and eight, calculate the t-statistic .
18. __________ Determine the p-value using the t-distribution.
19. __________ What is the largest confidence interval c for which this change is statistically significant?

Part Two

A Sanyo 256T washer forms a whirlpool in the wash tub during the wash cyle. The depth of the center of the whirlpool is proportional to the rotation rate of the water. The following data was recorded.

20. _________ Calculate the slope of the best fit (least squares) line for the data.
21. _________ Calculate the y-intercept of the best fit (least squares) line.
22. _________ Is the correlation positive, negative, or neutral?
23. _________ Use the equation of the best fit line to calculate the predicted depth for a rotation rate of 0.5 rotations per second.
24. _________ Use the inverse of the best fit line equation to calculate the rotation rate that would produce a whirlpool depth of 13 cm.
25. _________ Calculate the linear correlation coefficient r for the data.
26. _________ Is the correlation none, low, moderate, high, or perfect?
27. _________ Calculate the coefficient of determination.
28. _________ What percent of the variation in the rotation data explains the variation in the depth data?
29. _________ Is there a relationship between the rotation rate and the depth?

Basic Statistics
Statistic or Parameter	Symbol	Equations	Excel
Square root			=SQRT(number)
Sample size	n		=COUNT(data)
Sample mean	x	Sx/n	=AVERAGE(data)
Sample standard deviation	sx or s		=STDEV(data)
Sample Coefficient of Variation	CV	100(sx/x)	=100*STDEV(data)/AVERAGE(data)

Linear Regression Statistics
Statistic or Parameter	Symbol	Equations	Excel
Slope	b		=SLOPE(y data, x data)
Intercept	a		=INTERCEPT(y data, x data)
Correlation	r		=CORREL(y data, x data)
Coefficient of Determination	r2		=(CORREL(y data, x data))^2

Statistic or Parameter	Symbol	Equations	Excel
Normal Statistics
Calculate a z value from an x	z	=	=STANDARDIZE(x, µ, s)
Calculate an x value from a z	x	= s z + µ	=s*z+µ
Calculate a t-statistic (t-stat)	t		=(x - µ)/(sx/SQRT(n))
Calculate an x from a z			=µ + zc*sx/sqrt(n)
Find a probability p from a z value			=NORMSDIST(z)
Find a z value from a probability p			=NORMSINV(p)
Confidence interval statistics
Degrees of freedom	df	= n-1	=COUNT(data)-1
Find a zc value from a confidence level c	zc		=ABS(NORMSINV((1-c)/2))
Find a tc value from a confidence level c	tc		=TINV(1-c,df)
Calculate an error tolerance E of a mean for n >= 30 using sx	E		=zc*sx/SQRT(n)
Calculate an error tolerance E of a mean for n < 30 using sx. Can also be used for n >= 30.	E		=tc*sx/SQRT(n)
Calculate a confidence interval for a population mean µ from a sample mean x and an error tolerance E		x-E<= µ <=x+E
Hypothesis Testing
Calculate t-critical for a two-tailed test	tc		=TINV(a,df)
Calculate a p-value from a t-statistic	p		= TDIST(ABS(tstat),df,#tails)