U3L1: Basics of Hypothesis Testing (Using a Critical Value or P-value to Assess the Test Statistic)

procedure for finding P-values:

- What type of test is it? * left tailed * two-tailed * right-tailed - If left-tailed: * the P-value equals the area to the left of the test statistic (the line dividing the curve) - If two-tailed: * the P-value equals twice the area to the left of the test statistic * the P-value equals twice the area to the right of the test statistic - If right-tailed: - the P-value equals the area to the right of the test statistic (the line dividing the curve)

critical value

- any value that separates the critical region from the values of the test statistic that do not lead to rejection of the null hypothesis

finding a P-value:

- ex. consider the claim that the XSORT method of gender selection increases the likelihood of having a baby girl, so that p >0.5. Use the test statistic z = 3.21 (found from 13 girls in 14 births) and α = 0.05 - Step 1: Determine where the critical region falls: right tail. This means the P-value= area to the right of the test statistic - Step 2: Because the critical region is in the right tail, the P-value equals the area to the right of the test statistic value - Step 3: Use the given test statistic z=3.21 and Table A-2 chart to find the area to the left - Step 4: The area to the right is 1-0.9993=0.0007 - Step 5: Interpret P-value is 0.0007. Since 0.0007 is very small, it shows that there is a very small chance of getting the sample results that would led to a test statistic of z=3.21

finding the critical t-values for a two-tailed test:

- ex. n=12; α=0.05 Step 1: Use Table A-3 to find the critical t-values Step 2: Use "Area in Two Tails" section under 0.05 Step 3: Find df= n-1= 11 in the left column Step 4: Because there are two tails, their is a positive and negative value: t=±2.201

how to find critical values for a critical region (<):

- ex. using a significance level of α = 0.05, find the z value for the alternative hypotheses H_1: P< 0.5 Step 1: Recognize that p< 0.5 means that the critical region is in the left tail Step 2: Use Table A-2 to find the z-score that corresponds to the area to the left (0.05): z= -1.645 Step 3: The critical value is z=-1.645

if you know the cumulative area to the left and you want to find the critical z-value:

- good to use when the exact cumulative are to the left is not shown on the Table A-2 chart b/c it is between numbers 1. Press 2nd VARS 2. Press 3 for 3:invNorm( 3. Enter in the number for the cumulative area to the left 4. Enter and the result is the z-score

when to fail to reject the null hypothesis:

- if the P-value is normal or higher than 0.05 "if the P is high, the null will fly!"

when to reject the null hypothesis:

- if the P-value is very small, such as 0.05 or less - "if the P is low, the null must go"

critical/rejection region

- the set of all values of the test statistic that cause us to reject the null hypothesis - the critical regions of the test statistic are identified using the tail regions based on a significance level

P-value (probability value)

- the probability of getting a value of the test statistic that is at least as extreme as the one representing the sample data, assuming that the null hypothesis is true - P-value is NOT a population proportion 'p' - P-values are found AFTER finding the area beyond the test statistic

significance level (α)

- the probability that the test statistic will fall in the critical region when the null hypothesis is actually true or.. - the probability of making a mistake of rejecting the null hypothesis when it is actually true - used to calculate the critical value

how to find critical values for a critical region (≠):

- using a significance level of α = 0.05, find the two critical z values for the alternative hypotheses H_1: p≠ 0.5 Step 1: Recognize that p≠ 0.5 means that the critical region is in two tails. Step 2: If α = 0.05, then each of the two tails has an area of 0.025 Step 3: Using Table A-2, the left critical value is found by searching for the z-score of the cumulative left area of 0.025: z=-1.96 Step 4: Using Table A-2, the right most critical value is found by searching for the z-score of the cumulative left area of 0.975: z= 1.96

left-tailed test

- when the critical region is in the extreme left region (tail) under the curve - H_1 is <

right-tailed test

- when the critical region is in the extreme right region (tail) under the curve - H_1 is >

two-tailed test

- when the critical region is in the two extreme regions (tails) under the curve - α would be split between the two tails - ex. α=0.05 => α=0.025 per tail - H_1 is ≠

how to find critical values for a critical region (>):

ex. using a significance level of α = 0.05, find the z value for the alternative hypotheses H_1: p>0.5 Step 1: Recognize that p> 0.5 means that the critical region is in the right tail Step 2: If the right-tailed critical region is 0.05, the the cumulative area to the left is 0.95 Step 3: Use Table A-2 to find the z-score that corresponds to the area of 0.95: z= 1.645 Step 4: The critical value, the line that marks where the tail is on the curve, is z=1.645