Two Tailed Z-Test of Single Population Mean Hypothesis Testing

Two Tailed Z-Test of single population mean: Hypothesis testing • To statistically test if the population mean is equal to certain hypothetical value. • Assumptions – Population is normally distributed – If not normal, requires large samples – Population has known or good estimate of its standard deviation, thus Z-scores can be calculated Two Tailed Z-Test of population mean Case study: Tablets of a drug are compressed in a machine at target average weight 300 mg. A batch was made and a random sample of 30 tablets showed an average weight of 307. The company has specified s to be 15 mg. Does the batch meet target tablet weight at significance level (a ) of 0.05. Null hypothesis or the hypothesis to be tested (H0) – a claim that there is no difference between the population mean and the hypothesized value. H0: m = 300 368 gm. Alternative hypothesis HA– a statement of what we will believe is true if our sample data cause us to reject the null hypothesis. H1: m ¹ 300 At a=0.05 (95% confidence), the null hypothesis suggests the following distribution of the sample mean with m = m = 300 X s = SE = X s 15 = = 2.7 n 30 It suggests 95% of the possible samples should be within the confidence interval of the hypothetical mean (294.7-305.3) Of corresponding Z-scores of +1.96 and -1.96. Area above +1.96 = area below -1.96 =0.025 -The null hypothesis suggests that the mean tablet weight is not significantly different from 300 (not significantly higher or lower than 300). Thus, the test is two sided. Since a =0.05, it is split to two equal halves (a/2) as upper and lower tail areas of 0.025 each. Thus area below Z(1-a/2) or 1.96 = 0.975. Z values of 1.96 and -1.96 are referred to as critical values, because Z-score of the sample will be compared with these critical values. -If Z-score of the sample (Z-sample or test statistic) is higher than the upper critical value of 1.96 (in right tail region of a/2) or lower than the lower critical value of -1.96 (in the left tail region of a/2), we reject the null hypothesis and thus we can conclude that the population (never say sample mean) mean (mean tablet weight of the batch) is significantly different from 300 at a = 0.05. Thus left and right tails of a are called rejection regions (grey regions in the figure below) -On the other hand, if Z-sample is in between the critical values inclusive, we do not the null hypothesis and conclude that the population mean is not significantly different from 300 Test statistic (Z-sample) versus critical values of the case study _ zsample = teststatistic = X− m _ x s/ n 307 − 300 zsample = = 2 .6 2. 7 If Z-sample (test statistic) is between +1.96 and -1.96 inclusive (-1.96≤ Z-sample≤ +1.96), we do not reject the null hypothesis. But the calculated Z of the sample (2.6) was outside this range (≥+1.96), thus null hypothesis is rejected meaning that population mean (average tablet weight of the batch) is significantly different from 300 at significance level of 0.05 (always refer to a in your conclusions). A) B) According to H0, the population has a µ equal to its hypothesized value. If H0 is true and at a = 0.05, using this value, s and n, 95% of the samples drown from the population will give Zscores between -1.96 and +1.96. Thus it is very likely ( P 0.95) a randomly selected sample will have a Z-score in this critical range . If this happened, true H0 wont be rejected. Consequently, the decision is correct. HA suggest that the population mean is significantly different from that hypothesized by H0, and thus the population distribution is significantly shifted to right or to the left with respect to that suggested by H0. If HA is true, then the z-score of a randomly selected sample calculated using the hypothesized µ of H0 will likely be higher than +1.96 if the distribution is up shifted or less than -1.96 if the distribution is down shifted. With such sample, H0 will be correctly rejected. A) B) If H0 is true and HA is false, there is a low probability (P) of 0.05 (red areas of the two tails combined), that a randomly selected sample would give a Z-score higher than +1.96 (P 0.025) or lower than -1.96 (P 0.025). If this happens a true H0 will be rejected falsely. This called Type I error (a),which is to reject H0 when it is true. If HA is true and H0 is false, but the true distribution (HA) overlap with the inner critical region of the false distribution (the little blue area), a sample that is drawn by chance form this area gives a Z-score within the acceptance region of the false H0. In this case, a false H0 will be accepted and this is called Type II error. The type II error rate is often denoted as ,which is to fail to reject H0 when its false. Different critical values for different significance levels (a) As a is lower, it is less likely to reject a true H0, because the tail areas get smaller and vice versa If for the case study another batch was produced and the mean tablet weight of the sample was 295 mg perform Z-test at 0.1, 0.05 and 0.01 significance levels -UC and LC: upper and lower critical values, SM: sample mean,UCL and LCL: upper and lower confidence limits of sample mean. Reject H0 if LC≤ Z-sample≤ UC Common levels used in hypothesis testing; 0.05 is the most commonly used as a conventional level a UC LC 0.1 0.05 0.01 1.65 1.96 2.58 -1.65 -1.96 -2.58 Z-sample _ Z − sample = teststatistic = `X − m _ x s/ n Sample mean - Hypotheciz ed mean = SE 295 − 300 Z − sample = = −1.85 2. 7 Decision -1.85 (falls outside the critical range) Reject H0 -1.85 (falls inside the critical range) Fail to reject H0 -1.85 (falls inside the critical range) Fail to reject H0 -The lower the significance level (a) the wider the critical range. Thus the higher the chance not to reject H0. Conversly, the higher a, the higher the chance to reject H0 if it is true . P-value (two sided): – The p value of the hypothesis testing is the probability of obtaining, a value of the test statistic as extreme or more extreme (in the direction of supporting HA) than the one actually computed. – The p value indicates the probability of observing Z ≥ Z-sample or Z ≤ -Z-sample.In our case study: p value = P(Z ≥2.6) + P(Z≤2.6)= 0.0047+0.0047 or 2  0.0047 = 0.0094 If in the case study, 3 more batches were made and tested at the same sample size and a (0.05), but they had different sample means of 300, 297.3, 305.4 and 293. Calculate the p-value for each batch. When Z score of the sample mean is within the critical region (-1.95 to +1.96), pvalue is 0.05 or higher (Fail to reject H0 as test statistic is within the acceptance region) . When it is outside the critical region p-value is less than 0.05 (reject the H0 as test statistic is in the rejection region). At a = 0.05, the best case scenario and the worst case scenario for the acceptance of H0 are having Z scores of sample mean as 0 and 1.96, respectively. As Zsample higher deviates positively from the upper critical value or negatively from the lower critical value, p-value becomes smaller than 0.05 and thus the evidence against H0 becomes stronger. Conventions for interpreting P values at a = 0.05 (most commonly used) P Value Interpretation P > .05 Result is not significant; usually indicated by no asterisk P < .05 Result is significant; usually indicated by one asterisk (*) P < .01 Result is highly significant; usually indicated by two asterisks (**) P < .001 Result is very highly significant; usually indicated by three asterisks (***) If for the case study, 3 more batches were made and tested at the same sample size and a (0.05), but they had different sample means (SM) of 294, 292 and 290. The three batches showed different significant differences from the hypothesized population mean as indicated by asterisks for different p-values. Batch 1 2 3 SM 294 292 290 SE 2.7 2.7 2.7 Z-sample P-two sided -2.22 -2.96 -3.7 0.0264* 0.003** 0.0002*** At the same significance level, zcriticals are the same (1.96 for a=0.05), but samples of different means have different Z-scores (Zsample) and as these scores are more different from the critical values towrd the tails, lower pvalues are obtained. Making a Decision Decision Rule Based on P-value To use a P-value to make a conclusion in a hypothesis test, compare the P-value with a. 1. 2. If P a, then fail to reject H0. Claim Decision Claim is H0 Claim is Ha Reject H0 There is enough evidence to reject the claim. There is enough evidence to support the claim. Do not reject H0 There is not enough evidence to reject the claim. There is not enough evidence to support the claim. Testing H0 by Means of A Confidence Interval • When testing a null hypothesis by the mean of confidence interval, we reject H0 at the α level of significance (0.05 in the case study) if the hypothesized mean (suggested by H0) is not contained within the 95% confidence interval of the sample mean. • If the hypothesized mean is contained within the interval, H0 can not be rejected at α level of significance. • For the case study: The hypothesized s XZ * = parameter (300) is not n contained in the confidence 307  1.96  2.7 interval of the sample mean. Thus the null 301.7 to312.3 hypothesis is rejected 14 1 − a / 2