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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.
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Section 8.4Testing a claim about a mean(σ known) Objective For a population with mean µ (withσknown), 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)
(1) The population standard deviation σis known (2) One or both of the following: The population is normally distributed or n > 30 Requirements
Test Statistic Denoted z (as in z-score) since the test uses the z-distribution.
Example 1 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. What we know: µ0= 166.3 n= 40 x = 172.55σ = 26 Claim: µ> 166.3 using α= 0.1
Using Critical Region Example 1 What we know: µ0= 166.3 n= 40 x = 172.55σ = 26 Claim: µ> 166.3 using α= 0.05 H0:µ = 166.3 H1:µ > 166.3 Right-tailed Test statistic: zα = 1.282 z = 1.520 Critical value: z in critical region 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.
Calculating P-value for a Mean(σ known) Stat → Z statistics → One sample → with summary
Calculating P-value for a Mean(σ known) Then hit Calculate
Calculating P-value for a Mean(σ known) The resulting table shows both the test statistic (z) and the P-value P-value Test statistic P-value = 0.0642
Using P-value Example 1 What we know: µ0= 166.3 n= 40 x = 172.55σ = 26 Claim: µ> 166.3 using α= 0.05 H0:µ = 166.3 H1:µ > 166.3 Stat → Z statistics→ One sample → With summary ● Hypothesis Test Sample mean: Standard deviation: Sample size: 172.55 26 40 Null: proportion= Alternative 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.
Example 2 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. What we know: µ0= 250 n= 54 x = 237.9σ = 37.8 Claim: µ< 250 using α= 0.05
Using Critical Region Example 2 What we know: µ0= 250 n= 54 x = 237.9σ = 37.8 Claim: µ< 250 using α= 0.05 H0:µ = 250 H1:µ < 250 Left-tailed Test statistic: –zα = –1.645 z = –2.352 Critical value: z in critical region 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.
Using P-value Example 2 What we know: µ0= 250 n= 54 x = 237.9σ = 37.8 Claim: µ< 250 using α= 0.05 H0:µ = 250 H1:µ < 250 Stat → Z statistics→ One sample → With summary ● Hypothesis Test Sample mean: Standard deviation: Sample size: 237.9 37.8 54 Null: proportion= Alternative 250 < P-value = 0.0093 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.
