Section 8.5 Testing a claim about a mean ( σ unknown )

Section 8.5Testing a claim about a mean(σunknown) Objective For a population with mean µ (withσunknown), use a sample to test a claim about the mean. Testing a mean (when σ known) uses the t-distribution

Notation

(1) The population standard deviation σis unknown (2) One or both of the following: Requirements The population is normally distributed or The sample size n > 30

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

Example 1 People have died in boat accidents because an obsolete estimate of the mean weight (of 166.3 lb.) was used. A random sample of n = 40 men yielded the mean x = 172.55 lb. and standard deviation s = 26.33 lb.Do not assume the population standard deviation is known. Test the claim that men have a mean weight greater than 166.3 lb. using 90% confidence. What we know: µ0= 166.3 n= 40 x= 172.55 s= 26.33 Claim: µ> 166.3 usingα= 0.1 Note: Conditions for performing test are satisfied since n >30

Using Critical Regions Example 1 What we know: µ0= 166.3 n= 40 x= 172.55 s= 26.33 Claim: µ> 166.3 usingα= 0.1 H0:µ = 166.3 H1:µ > 166.3 right-tailed test Test statistic: tα= 1.304 t = 1.501 Critical value: (df = 39) tin critical region Initial Conclusion:Sincetin critical region, Reject H0 Final Conclusion: Accept the claimthatthe mean weight is greater than 166.3 lb.

Calculating P-value for a Mean(σ unknown) Stat → T statistics → One sample → with summary

Calculating P-value for a Mean(σ unknown) • Enter the Sample mean (x) • Sample std. dev. (s) • Sample size(n) Then hit Next

Calculating P-value for a Mean(σ unknown) Select Hypothesis Test Enter the Null:mean(µ0) Select Alternative(“<“, “>”, or “≠”) Then hit Calculate

Initial Conclusion Since P-value < α (α = 0.1), reject H0 Final Conclusion Accept the claim the mean weight greater than 166.3 Ib Calculating P-value for a Mean(σ unknown) The resulting table shows both the test statistic (t) and the P-value Test statistic (t) P-value

Using the P-value Example 1 What we know: µ0= 166.3 n= 40 x= 172.55 s= 26.33 Claim: µ> 166.3 usingα= 0.1 H0:µ = 166.3 H1:µ > 166.3 Stat → T statistics→ One sample → With summary ● Hypothesis Test Sample mean: Sample std. dev.: Sample size: 172.55 37.8 40 Null: proportion= Alternative 166.3 > Using StatCrunch P-value = 0.0707 Initial Conclusion:Since P-value < α, Reject H0 Final Conclusion: Accept the claimthatthe mean weight is greater than 166.3 lb.

P-Values A useful interpretation of the P-value: it is observed level of significance Thus, the value 1 – P-value is interpreted as observed level of confidence Recall: “Confidence Level” = 1 – “Significance Level” Note: Only useful if we reject H0 If H0 accepted, the observed significance and confidence are not useful.

P-Values From Example 1: P-value = 0.0707 1 – P-value= 0.9293 Thus, we can say conclude the following: The claim holds under 0.0707 significance. or equivalently… We are 92.93% confident the claim holds

Example 2 Loaded Die When a fair die (with equally likely outcomes 1-6) is rolled many times, the mean valued rolled should be 3.5 Your suspicious a die being used at a casino is loaded (that is, it’s mean is a value other than 3.5) You record the values for 100 rolls and end up with a mean of 3.87 and standard deviation 1.31 Using a confidence level of 99%, does the claim that the dice are loaded? What we know: µ0= 3.5 n= 100 x= 3.87 s= 1.31 Claim: µ ≠ 3.5 usingα= 0.01 Note: Conditions for performing test are satisfied since n >30

Using Critical Regions Example 2 What we know: µ0= 3.5 n= 100 x= 3.87 s= 1.31 Claim: µ ≠ 3.5 usingα= 0.01 H0:µ = 3.5 H1:µ≠ 3.5 two-tailed test Test statistic: zα = 2.626 zα = -2.626 z = 3.058 Critical value: (df = 99) tin critical region Initial Conclusion:Since P-value < α, Reject H0 Final Conclusion: Accept the claimthe die is loaded.

Using the P-value Example 2 What we know: µ0= 3.5 n= 100 x= 3.87 s= 1.31 Claim: µ ≠ 3.5 usingα= 0.01 H0:µ = 3.5 H1:µ≠ 3.5 Stat → T statistics→ One sample → With summary ● Hypothesis Test Sample mean: Sample std. dev.: Sample size: 3.87 1.31 100 Null: proportion= Alternative 3.5 ≠ Using StatCrunch P-value = 0.0057 Initial Conclusion:Since P-value < α, Reject H0 Final Conclusion: Accept the claimthe die is loaded. We are 99.43% confidence the die are loaded

Section 8.5 Testing a claim about a mean ( σ unknown )