Math 4030 – 9b Introduction to Hypothesis Testing

Example (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 36 new batteries and subject them to recharge cycles until they fail. The claim will be established if the sample mean life time is greater than 1660 recharge cycles. Otherwise, the claim will not be established and further improvements are needed.

Assuming  = 1600, sample mean distribution (from a sample of size 36) or  = 1600

P-value Approach:  = 1600 Sample mean (or size 36) distribution under the assumption  = 1600 P-Value = Probability of having such a “bad sample” or even worse.  = 1600 Sample mean > 1660

Null Hypothesis:  = 1600 (cycles) Alternative hypothesis:  > 1600 (cycles) (Researcher’s Claim) Level of significance:  = 0.05 Statistics: P-Value: Conclusion: the P-value is less than 0.05, the null hypothesis is rejected. We have enough evident to support the claim that the mean life of the new batteries is longer than 1600 cycles.

Basic Elements in Hypothesis Testing (P-Value Approach) Null hypothesis and Alternative hypothesis; Level of significance ; Tail(s) of the test; Sample statistic(s) and distribution(s); Calculation of P-value; Conclusion; Errors in Hypothesis Testing.

Null Hypothesis vs. Alternative Hypothesis The Null hypothesis, denoted by H0, is set up as an assumption that the distribution of the sample statistic(s) will be based on; The Alternative hypothesis, denoted by H1, is set up as an alternative assumption when the null hypothesis is declared false;

Level of Significance : Common choices for level of significance : 0.1, 0.05, 0.01, 0.001 Rules that  plays in the hypothesis testing; 1 - : confidence;  relate to probability of making certain error;

One-Tail vs. Two-Tails tests: Need for one tail test… Setting in the null and alternative hypothesis: Left Tailed or Right tailed? P-value in One-tail and Two-tail tests. Advantage of using one-tail test. What to watch for?

Sample statistics and distributions: Null hypothesis gives assumed values for population parameters; If the null hypothesis is true, then the sample statistic(s) should follow certain distribution; If the sample statistic is a value that seems unlikely to occur, we calculate this probability (P-Value) and compare it with the  value; If there this P-Value is smaller than , then the null hypothesis will be rejected.

Conclusion: If the null hypothesis is rejected, … If the null hypothesis is not rejected, … If the null hypothesis is research’s “intended” claim,…

Null Hypothesis:  = 1600 (cycles) Alternative hypothesis:  > 1600 (cycles) (Researcher’s Claim) Level of significance:  = 0.05 Statistics: Since we have a large sample, we may use CLT. By z-Table we find the P-Value: Conclusion: the P-value is less than 0.05, the null hypothesis is rejected. We have enough evident to support the claim that the mean life of the new batteries is longer than 1600 cycles.

Example (lithium car batteries): A research group is making great advances using a different type anode. They claim that the mean life of the new batteries is still 1600 recharge cycles. To verify this, they randomly select 36 new batteries and subject them to recharge cycles until they fail. The sample of 36 results a sample mean of 1660 cycles. Can the claim be confirmed at the significance level of  = 0.05?

Null Hypothesis:  = 1600 (cycles) (Researcher’s Claim) Alternative hypothesis:  ≥ 1600 (cycles) Level of significance:  = 0.05 (two-tailed) Statistics: Since we have a large sample, we may use CLT. By z-Table we find the P-Value: Conclusion: the P-value is great than 0.05, the null hypothesis is not rejected. We do not have enough evident to reject the claim that the mean life of the new batteries still 1600 cycles.

Errors in hypothesis testing: H0 is true H0 is false Reject H0 Type I error (Probability  ) No error Fail to reject H0 Type II error (Probability  )