1. 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)

2. 2 Notation

3. 3 (1) The population standard deviation σ is known (2) One or both of the following: The population is normally distributed or n > 30 Requirements

4. 4 Test Statistic Denoted z (as in z-score) since the test uses the z-distribution.

5. 5 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. Example 1 What we know: µ 0 = 166.3 n = 40 x = 172.55 σ = 26 Claim: µ > 166.3 using α = 0.1

6. 6 H 0 : µ = 166.3 H 1 : 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 H 0 Final Conclusion: We Accept the claim that the actual mean weight of men is greater than 166.3 lb. z = 1.520z α = 1.282 What we know: µ 0 = 166.3 n = 40 x = 172.55 σ = 26 Claim: µ > 166.3 using α = 0.05 Using Critical Region

7. 7 Stat → Z statistics → One sample → with summary Calculating P-value for a Mean (σ known)

8. 8

9. 9 Then hit Calculate Calculating P-value for a Mean (σ known)

10. 10 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)

11. 11 Using P-value Stat → Z statistics→ One sample → With summary Null: proportion= Alternative Sample mean: Standard deviation: Sample size: Example 1 ● Hypothesis Test 172.55 26 40 166.3 > P-value = 0.0642 Initial Conclusion: Since P-value < α, reject H 0 Final Conclusion: We Accept the claim that the actual mean weight of men is greater than 166.3 lb. H 0 : µ = 166.3 H 1 : µ > 166.3 What we know: µ 0 = 166.3 n = 40 x = 172.55 σ = 26 Claim: µ > 166.3 using α = 0.05

12. 12 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: µ 0 = 250 n = 54 x = 237.9 σ = 37.8 Claim: µ < 250 using α = 0.05

13. 13 H 0 : µ = 250 H 1 : µ < 250 Example 2 Left-tailed z in critical region Test statistic: Critical value: Initial Conclusion: Since z is in the critical region, reject H 0 Final Conclusion: We Accept the claim that the mean weight of bears is less than 250 lb. z = –2.352 –z α = –1.645 Using Critical Region What we know: µ 0 = 250 n = 54 x = 237.9 σ = 37.8 Claim: µ < 250 using α = 0.05

14. 14 Using P-value Null: proportion= Alternative Sample mean: Standard deviation: Sample size: Example 2 ● Hypothesis Test 237.9 37.8 54 250 < P-value = 0.0093 H 0 : µ = 250 H 1 : µ < 250 What we know: µ 0 = 250 n = 54 x = 237.9 σ = 37.8 Claim: µ < 250 using α = 0.05 Initial Conclusion: Since P-value < α, reject H 0 Final Conclusion: We Accept the claim that the mean weight of bears is less than 250 lb. Stat → Z statistics→ One sample → With summary