1. One sample t-test and z-test

2. Z-test / t-test
A test of whether the mean of a normally distributed population has a value specified in a null hypothesis. The z-test is used when - the population standard deviation is known or when n >3o. The t-test is used when the standard deviations are measured from the sample and n 30.

3. Z-test
The z value is based on the sampling distribution of X, which is normally distributed when the sample is reasonably large (recall Central Limit Theorem).
data points should be independent from each other
 the distributions should be normal if n is low, if however n>30 the distribution of the data does not have to be normal
 the variances of the samples should be the same
 all individuals must be selected at random from the population

4. Assumptions
normal distribution of data (e.g. Wilk-Shapiro normality test, KS-test)
equality of variances (F test, or more robust Levene's test)
Samples may be independent or dependent, depending on the hypothesis and the type of samples:
Independent samples are usually two, randomly selected groups
Dependent samples are either two groups matched on some variable (for example, age) or are the same people being tested twice (called repeated measures)

5. One-Sample z-test
Test for the population mean from a large sample with population standard deviation known

6. Example 1
The current rate for producing 5 amp fuses at Neary Electric Co. is 250 per hour. A new machine has been purchased and installed that, according to the supplier, will increase the production rate. The production hours are normally distributed. A sample of 35 randomly selected hours from last month revealed :the following measures:
At the .05 significance level can Neary conclude that the new machine is faster?

7. State the null and alternate hypotheses.
Step 1
State the null and alternate hypotheses.
H0: µ = 250 : Production rate is not significantly different from 250 per hour.
H1: µ > 250 : Production rate is significantly higher than 250 per hour.
Step 2
Select the level of significance.
= .05.
Step 3
Find a test statistic.
Use the z distribution since s is not known but n > 30.
Step 4
State the decision rule.
Reject the null hypothesis if z > or, using the p-value, the null hypothesis is rejected if p < .05.

8. Reject Ho Step 5 Calculate the value of the test statistics Step 6
Make a decision and interpret the results.
Since Computed z of 5.72 > Critical z of
Reject Ho
Step 7
State the conclusion.
The mean number of fuses produced is significantly higher than 250 per hour.

9. One-Sample t-test
Test for the population mean from a small sample with population standard deviation unknown

10. Example 2:
The US Farmers’ Production Company builds large harvesters. For a harvester to be properly balanced, a 25-pound plate is installed on its side. The machine that produces these plates is set to yield plates that average 25 pounds. The distribution of plates produced from the machine is normal. However, the shop supervisor is worried that the machine is out of adjustment and is producing plates that do not average 25 pounds. To test this concern, he randomly selects 20 of the plates produced the day before and weighs them. The results are shown in the next slide.
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 21

11. Weights in Pounds of a Sample of 20 Plates
Test if the average weight is equal to 25 pounds.

12. Two-tailed Test: Small Sample,  Unknown,  = .05
Critical Values
Non Rejection Region
Rejection Regions
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 22

13. Two-tailed Test: Small Sample,  Unknown,  = .05 (part 3)
Critical Values
Non Rejection Region
Rejection Regions
Business Statistics: Contemporary Decision Making, 3e, by Black. © 2001 South-Western/Thomson Learning 23

14. exercises
1. The environmental Protection Agency releases figures on urban air soot in selected cities in the United States. For the city of St. Louis, the EPA claims that the average number of micrograms of suspended particles per cubic meter of air is 82. Suppose St. Louis officials have been working with businesses, commuters, and industries to reduce this figure. These city officials hire an environmental company to take random measures of air soot over a period of several weeks. The resulting data follow. Use these data to determine whether the urban air soot in St. Louis is significantly lower than it was when the EPA conducted its measurements. Let =.01.

15. 81.6
66.6
70.9
82.5
58.3
71.6
72.4
96.6
78.6
76.1
80.0
73.2
85.5
68.6
74.0
68.7
83.0
86.9
94.9
75.6
77.3
86.6
71.7
88.5 87
72.5
85.8
74.9
61.7
92.2