Exercises from the E Text
Patrice White
RES/341
July 19, 2011
Larry Myers
University of Phoenix

8.46 A random sample of 10 miniature Tootsie Rolls was taken from a bag. Each piece was weighed on a very accurate scale. The results in grams were
3.087 3.131 3.241 3.241 3.270 3.353 3.400 3.411 3.437 3.477
(a) Construct a 90 percent confidence interval for the true mean weight. (b) What sample size would be necessary to estimate the true weight with an error of } 0.03 grams with 90 percent confidence?
(c) Discuss the factors which might cause variation in the weight of Tootsie Rolls during manufacture. (Data are from a project by MBA student Henry Scussel.) Tootsie
Answers:
a) Confidence intervals are used to find a region in which we are 100 * ( 1 - α )% confident the true value of the parameter is in the interval.
In order for the Confidence Interval to be valid you must have data from a normal distribution, at least if you are using the method here. If you do not have normal data then this type of confidence interval is not valid.

To clear up the notation I will use here. "t" is the test statistic and "t_(n-1)" is a Student t random variable with n - 1 degrees of freedom, e.g. a Student t random variable with 18 degrees of freedom is denoted as t_18.For small sample confidence intervals about the mean you have: xBar ± t * sx / sqrt(n)

where xBar is the sample mean t is the t - score with n - 1 degrees of freedom such that α% of the data in the tails, i.e., P( |t_(n-1)| > t) = α sx is the sample standard deviation n is the sample size

The sample mean xbar = 3.3048
The sample standard deviation sx = 0.1319889
The sample size n = 10

The t score for a 0.9 confidence interval is the t score such that 0.05 is in each tail. t = 1.833113
The confidence interval is:

( xbar - t * sx / sqrt( n ) , xbar + t * sx / sqrt( n ) )
( 3.228289 , 3.381311 )

b)

Confidence intervals are used to find a region in which we are 100 * ( 1 - α )% confident the true value of the parameter is in the interval.

For large sample confidence intervals about the mean you have:

xBar ± z * sx / sqrt(n)

where xBar is the sample mean z is the zscore for having α% of the data in the tails, i.e., P( |Z| > z) = α sx is the sample standard deviation n is the sample size

To find the sample size needed for a confidence interval of a given size we need only to concern ourselves with the error term of the CI.
We know that the interval is centered at xbar so we need to find the value of n such that

z * sx / sqrt(n) = width.

The z-score for a 0.9 confidence interval is the value of z such that 0.05 is in each tail of the distribution. z= 1.644854

The equation we need to solve is: z * sx / sqrt(n) = width

n = (z * sx / width) ^ 2.

n = ( 1.644854 * 0.1319889 / 0.03 ) ^ 2 n = 52.3705

Since n must be integer valued we need to take the ceiling of this solution. Always take the ceiling so that the size of the CI will be correct.

n = 53

c) the mixture of the dough, the measuring machines, humidity in the factory, etc....

8.62 In 1992, the FAA conducted 86,991 pre-employment drug tests on job applicants who were to be engaged in safety and security-related jobs, and found that 1,143 were positive. (a) Construct a
95 percent confidence interval for the population proportion of positive drug tests. (b) Why is the normality assumption not a problem, despite the very small value of p? (Data are from Flying 120, no. 11 [November 1993], p. 31.)
Answer:
a) This is a binomial problem and so I used the formula

θ(hat) +/- 1.96(sqrt[θ(hat){1-θ(hat)}]/n)

θ(hat) is the proportion of successes, in this case 1,143/86,991 = 0.013 [note 1 - 0.013 = 0.987]

so CI is 0.013 +/- 1.96(sqrt(0.013*0.987)/86,991)
= 0.013 +/- 0.000753

CI is (0.0122, 0.0138)

5. State the main points of the Central Limit Theorem for a mean.
Answer:
(1) Random samples can be drawn from any population, normally distributed or not. Thus, even if it is known that the dollar value of a certain inventory item is not normally distributed, the theorem can be invoked and the assumption made that the sample mean inventory dollar value will be normally distributed.

(2) The theorem allows statements to be made about the value of the population mean without looking at the entire population. Thus, interval estimates can be made about the true value of an inventory item. Such interval estimates are called Confidence Interval.

6. Why is population shape of concern when estimating a mean? What does sample size have to do with it?
The greater the sample size, the less the probability that what is observed is due to chance. However, you get into diminishing returns, cost, and inconvenience if the sample is too large. The smallest that will show statistical significance with reasonable accuracy is about 25.