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 Notation

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

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

6 H 0 : µ = H 1 : µ > 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 lb. z = 1.520z α = What we know: µ 0 = n = 40 x = σ = 26 Claim: µ > using α = 0.05 Using Critical Region

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

8

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

10 The resulting table shows both the test statistic (z) and the P-value Test statistic P-value P-value = Calculating P-value for a Mean (σ known)

11 Using P-value Stat → Z statistics→ One sample → With summary Null: proportion= Alternative Sample mean: Standard deviation: Sample size: Example 1 ● Hypothesis Test > P-value = 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 lb. H 0 : µ = H 1 : µ > What we know: µ 0 = n = 40 x = σ = 26 Claim: µ > using α = 0.05

12 Weight of Bears A sample of 54 bears has a mean weight of 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 = σ = 37.8 Claim: µ < 250 using α = 0.05

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 = σ = 37.8 Claim: µ < 250 using α = 0.05

14 Using P-value Null: proportion= Alternative Sample mean: Standard deviation: Sample size: Example 2 ● Hypothesis Test < P-value = H 0 : µ = 250 H 1 : µ < 250 What we know: µ 0 = 250 n = 54 x = 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