1 tests of hypotheses about the mean - continued
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Tests of Hypotheses
about the mean
- continued
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Reminder
Two types of hypotheses:
H0 - the null hypothesis (e.g. μ=24)
H1 - the alternative hypothesis (e.g. μ>24)
Test statistic:
P-value: probability of obtaining values as extreme as or more extreme than the test statistic
e.g., P(Z≤-2)=0.0228
Decision at the α significance level:
Reject H0 if p-value<α
n
XZ
2
1003
244.23 :e.g.
Z
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Testing hypotheses using a confidence interval:
Example:A certain maintenance medication is supposed to contain a mean of 245 ppm of a particular chemical. If the concentration is too low, the medication may not be effective; if it is too high, there may be serious side effects. The manufacturer takes a random sample of 25 portions and finds the mean to be 247 ppm. Assume concentrations to be normal with a standard deviation of 5 ppm. Is there evidence that concentrations differ significantly (α=5%) from the target level of 245 ppm?
Hypotheses:
H0: μ=245
H1: μ≠245
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First, lets examine the Z test statistic:
Test statistic:
P-value:
2P(Z>2)=2(0.0228)=0.0456
Decision at 5% significance level:
P-value>α reject H0
The concentration differs from 245
n
XZ
2
255
245247
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Now, examine the hypotheses using a confidence interval
α =0.05 confidence level is 1- α = 95%
95% CI:
[245.04 , 248.96]
We are 95% certain that the mean concentration is between 245.04 and 248.96.
Since 245 is outside this CI - reject H0.
The concentration differs from 245
n
zX
975. 25
596.1247
)1(96.1247
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H0: μ=μ0
H1: μ≠μ0
If μ0 is outside the confidence interval, then we reject the null hypothesis at the α significance level.
Note: this method is good for testing two-sided hypotheses only
[ confidence interval]
Examine the hypotheses using a confidence interval
μ0
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ExampleSuppose a claim is made that the mean weight μ for a population of male runners is 57.5 kg. A random sample of size 24 yields . [σ is known to be 5 kg].
Based on this, test the following hypotheses:
H0: μ=57.5
H1: μ≠57.5
Answer using:
a) A Z test statistic
b) A confidence interval
60X
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a)
Test statistic:
P-value:
2P(Z>2.45)=2(1-.9929)=2(.0071)=.0142
Decision at 5% significance level:
P-value<α reject H0
Conclusion:
Mean weight differs from 57.5
n
XZ
45.2
245
5.5760
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b) α =0.05 confidence level is 1- α = 95%
95% CI:
[58 , 62]
57.5 is outside this CI - reject H0.
Mean weight differs from 57.5
question?: Would you reject H0: μ=59 vs. H1: μ≠59?
No, because 59 is in the interval [58, 62]
n
zX
975. 24
596.160
)02.1(96.160
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Testing hypotheses using Minitab
In a certain university, the average grade in statistics courses is 80, and σ=11. A teacher at that university wanted to examine whether her students received higher grades than the rest of the stat classes. She took a sample of 30 students and recorded their grades
hypotheses:H0:μ=80H1:μ>80
data are:
mean: 03.85X
95 100 82 76 75 83 75 96 75 98 79 80 79 75 100 91 81 78 100 72 94 80 87 100 97 91 70 89 99 54
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Test statistic:
P-value:P(Z>2.51)=1-0.9940=0.006
Decision at 5% significance level:P-value<α reject H0
conclusion:The grades are higher than 80
n
XZ
51.2
3011
8003.85
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Minitab…
.\test of hypotheses.MPJ
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Column of scores:
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Choose:
Stat > Basic Statistics >1-Sample Z
Pick options
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In the “options” window pick the alternative hypothesis
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One-Sample Z: scores
Test of mu = 80 vs mu > 80The assumed sigma = 11
Variable N Mean StDev SE Meanscores 30 85.03 11.51 2.01
Variable 95.0% Lower Bound Z Pscores 81.73 2.51 0.006
In the session window:
Test statistic: Z = 2.51
P-value: 0.006
Decision: reject H0 at the 5% significance level
conclusion:The grades are higher than 80
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Use Minitab to build a confidence interval
Choose:
Stat > Basic Statistics >1-Sample Z
Pick options
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In the “options” window pick the two-sided alternative hypothesis
Pick “not equal”
Because a confidence interval is like a two sided hypothesis
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In the session window:
One-Sample Z: scores
Test of mu = 80 vs mu not = 80The assumed sigma = 11
Variable N Mean StDev SE Mean scores 30 85.03 11.51 2.01
Variable 95.0% CI Z Pscores ( 81.10, 88.97) 2.51 0.012
A 95% CI: [81.10, 88.97]
Equivalent to testing hypotheses:H0:μ=80H1:μ≠80
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Questions
1. Suppose H0 was rejected at α=.05. Answer the following questions as: “Yes”, “No”, “Cannot tell”:
(a) Would H0 also be rejected at α=.03?
(b) Would H0 also be rejected at α=.08?
(c) Is the p-value larger than .05?
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2. Suppose H0 was not rejected at α=.05. Answer the following questions as: “Yes”, “No”, “Cannot tell”:
(a) Would H0 be rejected at α=.03?
(b) Would H0 be rejected at α=.08?
(c) Is the p-value larger than .05?
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3. A 95% confidence interval for the mean time (in hours) to complete an audit task is: [7.04,7.76].Use the relation between confidence intervals and two-sided tests to examine the following sets of hypotheses:
(a) H0: μ=7.5 H1: μ≠7.5 (α=.05)
(b) H0: μ=7 H1: μ≠7 (α=.05)
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4. A 90% confidence interval for the mean is: [20.1, 23.5].We can use the relation between confidence intervals and two-sided tests to examine hypotheses about the mean
At what level of significance, α, can we test these hypotheses based on the confidence interval?
(a) α=.01(b) α=.025(c) α=.05(d) α=.1
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5. A 90% confidence interval for the mean is: [20.1, 23.5] has been used for testing the following hypotheses:
H0: μ=19 H1: μ≠ 19
H0 is rejected at 10% significance level(19 is outside the CI)
At what level of significance, α, can we still reject H0: μ=19?
Answer:
We can reject H0 for α<0.1