Math 4030 – 10a Tests for Population Mean(s) Z-test vs t-Test (in tests regarding one mean) Dependent and independent samples Comparing two means

Example 1 (lithium car batteries): A research group is making great advances using a new type anode. They claim that the mean life of the new batteries is greater than 1600 recharge cycles. To support this claim, they randomly select 16 new batteries and subject them to recharge cycles until they fail. This sample results in a sample mean of 1680 and a sample standard deviation of 208. Can this result support the claim that the new batteries is greater than 1600 recharge cycles? (Use  = 0.05) Assume that the population is normally distributed.

For small samples (n < 30), if the population is normal or near normal with unknown variance, then

P-value Approach: P-Value t Sample mean (or size 16) distribution under the assumption  = 1600 P-Value t T-score 1.54

Critical Region Method Null Hypothesis:  = 1600 (cycles) Alternative hypothesis:  > 1600 (cycles) (Researcher’s Claim) Level of significance:  = 0.05 (Right-tailed test) Statistics: Since we have a small sample from a normal population with unknown variance, we use t-test with df = 15. We find the Critical Value 1.753 and the Critical Region (1.753, ). The sample statistic is Conclusion: the sample statistic t-score does not fall in the critical region, we do not have enough evidence to reject the null hypothesis. …

T-test or z-test? P-Value or Critical Region Method? Size of samples Normality (population distribution) Population variance P-Value or Critical Region Method? Use of Tables or Computer? Fixed  level? Need of P-value?

Data format for independent samples: Population 1 (may or may not be normally distributed), with mean 1 (to be estimated and compared) and variance 21 (may or may not known). Sample of size n1: With sample mean and sample variance: Population 2 (may or may not be normally distributed), with mean 2 (to be estimated and compared) and variance 22 (may or may not known). Sample of size n2: With sample mean and sample variance:

Sampling distribution of : Distribution? CLT still apply?

Z-test can be used if If both populations are normal with known variances, then the mean difference is normally distributed; or If both samples are large (n1 ≥ 30, n2 ≥ 30). (Unknown population variances 2’s can be estimated by sample variances s2’s .) or

Confidence interval for Sample Statistic for or Confidence interval for or

Example 2 (Two independent samples): It is believed that the resistance of certain electric wire can be reduced by 0.05 ohm by alloying. (Assuming standard deviation of resistance of any wire is 0.035 ohm.) A sample of 32 standard wires and 32 alloyed wires are sampled. Question 1: Find the probability that average resistance of 32 standard wires is at least 0.03 ohm higher than that of 32 alloyed wires.

It is believed that the resistance of certain electric wire can be reduced by alloying. To verify this, a sample of 32 standard wires results the sample mean 0.136 ohm and sample sd 0.034 ohm, and a sample of 32 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.036 ohm Question 2: Construct a 95% confidence interval for the mean resistance reduction due to alloying.

It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 32 standard wires results the sample mean 0.136 ohm and sample sd 0.004 ohm, and a sample of 32 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.005 ohm. Question 3: Can we support the claim at  = 0.05 level?

When the samples are small, and the populations are normally distributed with unknown means, we need to use t-test. Degree of freedom? Pooled variance Two cases: Two populations are believed to have the same variance. Two populations are believed to have different variances.

Case 1: When the samples are small, and the populations are normally distributed with an unknown but equal variance.

Case 2: Small sample(s), normal populations with unequal variance has t distribution with estimated degree of freedom

Example 2’: It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 standard wires results the sample mean 0.136 ohm and sample sd 0.0049, and a sample of 15 alloyed wires results the sample mean of 0.083 ohm and sample sd 0.0052 . (Assume that the resistance has normal distribution with the same variance) Question: Can we support the claim at  = 0.05 level?

Dependent/Matched/Paired Samples: Only one population, and one sample of size n, but two measurements: Since we are interested in the differences, this is really a one sample problem: where

Sampling distribution of : Test the hypothesis D = 0 vs. Confidence interval containing 0.

Example 3: It is claimed that the resistance of certain electric wire can be reduced by more than 0.05 ohm by alloying. To verify this, a sample of 15 wires are tested before the alloying and again after the alloying, we find the mean reduction 0.063 ohm, and the sd of the reductions 0.025. (Assume that the resistance has normal distribution) Question: Can we support the claim at  = 0.05 level?