Section 8.4 Testing a claim about a mean ( σ known)
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1 Section 8.4 Testing a claim about a mean (σ known) Objective For a population with mean µ (with σ known), use a sample (with a sample mean) to test a claim about the mean. Testing a mean (when σ known) uses the standard normal distribution (z-distribution)
description
Section 8.4 Testing a claim about a mean ( σ known). Objective For a population with mean µ (with σ known ), use a sample (with a sample mean) to test a claim about the mean. Testing a mean (when σ known) uses the standard normal distribution ( z -distribution). Notation. - PowerPoint PPT Presentation
Transcript of Section 8.4 Testing a claim about a mean ( σ known)
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Section 8.4Testing a claim about a mean
(σ known)
ObjectiveFor a population with mean µ (with σ known), use a sample (with a sample mean) to test a claim about the mean.
Testing a mean (when σ known) uses the standard normal distribution (z-distribution)
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Notation
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(1) The population standard deviation σ is known
(2) One or both of the following:The population is normally distributed orn > 30
Requirements
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Test StatisticDenoted z (as in z-score) since the test uses the z-distribution.
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People have died in boat accidents because an obsolete estimate of the mean weight of men (166.3 lb.) was used.
A random sample of n = 40 men yielded the mean = 172.55 lb. Research from other sources suggests that the population of weights of men has a standard deviation given by = 26 lb.
Use a 0.1 significance level to test the claim that men have a mean weight greater than 166.3 lb.
x
Example 1
What we know: µ0 = 166.3 n = 40 x = 172.55 σ = 26
Claim: µ > 166.3 using α = 0.1
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H0 : µ = 166.3
H1 : µ > 166.3
Example 1
Right-tailed
z in critical region
Test statistic:
Critical value:
Initial Conclusion: Since z is in the critical region, reject H0
Final Conclusion: We Accept the claim that the actual mean weight of men is greater than 166.3 lb.
z = 1.520zα = 1.282
What we know: µ0 = 166.3 n = 40 x = 172.55 σ = 26
Claim: µ > 166.3 using α = 0.05
Using Critical Region
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Stat → Z statistics → One sample → with summary
Calculating P-value for a Mean(σ known)
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Calculating P-value for a Mean(σ known)
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Then hit Calculate
Calculating P-value for a Mean(σ known)
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The resulting table shows both the test statistic (z) and the P-value
Test statistic
P-value
P-value = 0.0642
Calculating P-value for a Mean(σ known)
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Using P-value
Stat → Z statistics→ One sample → With summary
Null: proportion=Alternative
Sample mean:Standard deviation:
Sample size:
Example 1
● Hypothesis Test172.55
26
40
166.3
>
P-value = 0.0642
Initial Conclusion: Since P-value < α, reject H0
Final Conclusion: We Accept the claim that the actual mean weight of men is greater than 166.3 lb.
H0 : µ = 166.3
H1 : µ > 166.3
What we know: µ0 = 166.3 n = 40 x = 172.55 σ = 26
Claim: µ > 166.3 using α = 0.05
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Weight of Bears
A sample of 54 bears has a mean weight of 237.9 lb.
Assuming that σ is known to be 37.8 lb. use a 0.05 significance level to test the claim that the population mean of all such bear weights is less than 250 lb.
Example 2
What we know: µ0 = 250 n = 54 x = 237.9 σ = 37.8
Claim: µ < 250 using α = 0.05
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H0 : µ = 250
H1 : µ < 250
Example 2
Left-tailed
z in critical region
Test statistic:
Critical value:
Initial Conclusion: Since z is in the critical region, reject H0
Final Conclusion: We Accept the claim that the mean weight
of bears is less than 250 lb.
z = –2.352
–zα = –1.645
Using Critical RegionWhat we know: µ0 = 250 n = 54 x = 237.9 σ = 37.8
Claim: µ < 250 using α = 0.05
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Using P-value
Null: proportion=Alternative
Sample mean:Standard deviation:
Sample size:
Example 2
● Hypothesis Test237.9
37.8
54
250
<
P-value = 0.0093
H0 : µ = 250
H1 : µ < 250
What we know: µ0 = 250 n = 54 x = 237.9 σ = 37.8
Claim: µ < 250 using α = 0.05
Initial Conclusion: Since P-value < α, reject H0
Final Conclusion: We Accept the claim that the mean weight
of bears is less than 250 lb.
Stat → Z statistics→ One sample → With summary
