Review 2 92.1.9
Chapter 7:
Compute the standard error and the probability of the sample mean or the sample proportion within some range.
Different sampling methods.
Example 1:
MNM Corporation gives each of its employees an aptitude test. The scores on the test are normally distributed with a mean of 75 and a standard deviation of 15. A simple random sample of 25 is taken from a population of 500.
(a)What is the probability that the average aptitude test score in the sample will be between 70.14 and 82.14?
(b)What is the probability that the average aptitude test score in the sample will be greater than 82.68?
(c)Find a value, C, such that P( C) = .015.
[solution:]
(a)
Since ,
(b)
(c)
Example 2:
A new soft drink is being market tested. It is estimated that 60% of consumers will like the new drink. A sample of 96 taste-tested the new drink.
(a) Determine the standard error of the proportion
(b)What is the probability that more than 70.4% of consumers will indicate they like the drink?
(c) What is the probability that more than 30% of consumers will indicate they do not like the drink?
[solution:]
(a)
(since
(b)
(c)
We need to compute the probability that less than or equal to 70% of consumers will indicate they like the drink?
Example 3:
Suppose we have a population of 40 elements
148 / 148 / 149 / 149 / 153 / 154 / 155 / 155 / 156 / 156157 / 157 / 158 / 158 / 158 / 158 / 158 / 159 / 159 / 160
160 / 160 / 161 / 162 / 162 / 162 / 163 / 163 / 163 / 163
164 / 164 / 164 / 164 / 165 / 165 / 165 / 165 / 165 / 166
Suppose the first row of the table of random number is
63271 59986 71744 51102 15141 80714 58683 93108 13554 79945
Please use systematic sampling to obtain
(a)a sample of 5 elements.
(b)the sample mean and sample variance based on (a).
[solution:]
(a)
40/5=8. Thus, we need to divide the original data into 5 subsets and select 1 element from these subsets. The subsets are
Subset1 / 148 / 148 / 149 / 149 / 153 / 154 / 155 / 155
Subset 2 / 156 / 156 / 157 / 157 / 158 / 158 / 158 / 158
Subset
3 / 158 / 159 / 159 / 160 / 160 / 160 / 161 / 162
Subset
4 / 162 / 162 / 163 / 163 / 163 / 163 / 164 / 164
Subset
5 / 164 / 164 / 165 / 165 / 165 / 165 / 165 / 166
The first random number between 1 and 8 are 6. Therefore, the sample we select are 154, 158, 160, 163, and 165.
(b)
The sample mean is
and the sample variance is
Chapter 8:
Construct a confidence intervals in large and small sample cases.
Determine sample size based on the desired margin of error
Example:
- A random sample of 81 workers at a company showed that they work an average of 100 hours per month with a standard deviation of 27 hours.
(a)Compute a 95% confidence interval for the mean hours per month all workers at the company work.
(b) At 95% confidence, how many more workers need to be included in the sample to provide a confidence interval with length 4 (i.e., the margin of error being 2)?
[solution:]
(a)
As ,
is a 95% confidence interval estimate of the population mean .
(b)
Since ,
.
Thus, we need 701-81=620 more workers.
2. For a t distribution with 16 degrees of freedom, find the area of probability.
(a) To the left of -1.746.
(b) Between -1.337 and 2.120.
[solution:]
(a)
.
(b)
1