Transcript 1 sample z or 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.
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
3
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)
4
Test for the population mean from
a large sample with population
standard deviation known
X 
z
/ n
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:
251
249
255
252
250
248
249
253
254
251
246
259
260
259
258
256
258
260
261
262
261
250
249
249
247
251
252
255
256
256
255
260
260
261
259
At the .05 significance level can
Neary conclude that the new
machine is faster?
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  is not
known but n > 30.
Step 4
State the decision rule.
Reject the null hypothesis if z > 1.645 or,
using the p-value, the null hypothesis is
rejected if p < .05.
Step 5
Calculate the value of the test statistics
z
X 

n

254 .63  250
4.79
35
 5.72
Step 6
Make a decision and interpret the results.
oSince Computed z of 5.72 > Critical z of 1.645
Reject Ho
Step 7
State the conclusion.
The mean number of fuses produced is
significantly higher than 250 per hour.
Test for the population mean from
a small sample with population
standard deviation unknown
t
X 
s/ n
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.
9-10
Business Statistics:
Contemporary Decision Making,
3e, by Black. © 2001 SouthWestern/Thomson Learning
Weights in Pounds of a Sample of 20 Plates
22.6
27.0
26.2
25.8
22.2
26.6
25.3
30.4
23.2
28.1
23.1
28.6
27.4
26.9
24.2
23.5
24.5
24.9
26.1
23.6
Test if the average weight is equal to 25 pounds.
X  25.51, S = 2.1933, and n = 20
Ho:   25
Ha:   25
Rejection Regions

2
df  n  1  19

.025
2
.025
Non Rejection Region
t
c
t
 2.093
Critical Values
9-12
Business Statistics:
Contemporary Decision Making,
3e, by Black. © 2001 SouthWestern/Thomson Learning
c
 2.093
If t  2.093, reject Ho.
Rejection Regions
If t  2.093, do not reject Ho.

.025
2

.025
2
Non Rejection Region
t
c
t
 2.093
c
 2.093
Critical Values
X   2551
.  250
.
t

 104
.
S
21933
.
n
20
Since t  104
.  2.093, do not reject Ho.
9-13
Business Statistics:
Contemporary Decision Making,
3e, by Black. © 2001 SouthWestern/Thomson Learning
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.
81.6
96.6
68.6
77.3
85.8
66.6
78.6
74.0
86.6
74.9
70.9
76.1
68.7
71.7
61.7
82.5
80.0
83.0
88.5
92.2
58.3
73.2
86.9
87
71.6
85.5
94.9
72.5
72.4
73.2
75.6
83.0