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O a Let Us Study To be able to answer the questions in the next activities, please take time to read and understand this section that discusses the next steps in hypothesis testing. Critical Value, Signicance Level, and Rejection Region In hypothesis testing, a critical value is a point on the test distribution that is compared to the test statistic to determine whether to reject the null hypothesis. Critical values for a test of hypothesis depend upon the test statistic, which is specic to the type of the test and signicance level (a) which denes the sensitivity of the test. A value of (120.05 implies that the null hypothesis is rejected 5% of the time when it is in fact true. In practice, the common values of a are 0.1, 0.05, and 0.01. Critical Value of z-Distribution A critical value of z (Z-score) is used when the sampling distribution is normal or close to normal. Zscores are used when the population standard deviation is known or when you have larger sample sizes. While the zscore can also be used to calculate probability for unknown standard deviations and small samples. Many statisticians prefer using the t distribution to calculate these probabilities. Table of Critical Values (ZScore) Level of Signicance 'r t es Type a: 0.01 left-tailed test \\ -2.33 right-tailed test 2.33 575 ms a. left-tailed test: If the alternative hypothesis Ha contains the lessthan inequality symbol (), the hypothesis test is a righttailed test. c. two-tailed test: If the alternative hypothesis Ha contains the notequal to symbol (), the hypothesis test is a twotailed test. In a twotailed test, each tail has an area of a. Examples: Find the critical 2 values based on the table of critical value of z. In each case, assume that the normal distribution applies. 1. lefttailed test with a: 0.01 2. twotailed test with d=0.05 3. righttailed test with c1=0.025 Answers: 1. z = 2.33 2. z = i196 3. z = 1.96 Critical Value of t-Distribution The t-distribution table values are critical values of the t- distribution. The column header is the tdistribution probabilities (alpha). The row names are the degrees of freedom (df). To nd critical values for tdistribution: 1. Identify the level of signicance. 2. Identify the degrees of freedom, d.f. = n l. 3. Find the critical value using tdistribution in the row with n1 degrees of freedom. If the hypothesis test is: a. left-tailed, use \"a one tail\" column with a negative sign. b. right-tailed, use \"(1 one tail\" column with a positive sign. c. two-tailed, use \"a two tails\" column with a negative and a positive sign. Examples: a) Find the critical tvalue for a lefttailed test with d= 0.05 and n =21. Answer: t:1.725 b) Find the critical tvalue for a righttailed test with (1=0.01 and n = 17. Answer: t = 2.583 c) Find the critical tvalues for a twotailed test with d=0.05 and n =26. Answer: t = i2.060 Critical Value Table for t-Distribution a for two-tailed test 0. 10 0.05 0.02 0.01 1 6.311 12.706 31.821 63.657 ' 2 2.920 4.303 6.065 9.925 3 2.353 3.182 4.541 5.841 ' 4 2.132 2.776 3.747 4.604 5 2.025 2.571 3.365 4.032 ' 6 1.943 2.447 3.143 3.707 '7 1.895 2.365 2.998 3.499 ' 8 1.860 2.306 2.896 3.355 9 1.833 2.262 2.821 3.250 ' 10' 1.812 2.228 2.764 3.169 11 1.796 2.201 2.718 3.106 ' 12 1.782 2.179 2.681 3.055 13 1.771 2.160 2.650 3.012 ' 14- 1.761 2.145 2.624 2.977 15 1.753 2.134 2.602 2.947 ' 16 1.746 2.120 2.583 2.921 17 1.740 2.110 2.567 2.898 13 1.734 2.101 2.552 2.878 19 1.729 2.093 2.539 2.861 20 1.725 2.086 2.528 2.845 21 1.721 2.080 2.512 2.831 22 1.717 2.074 2.508 2.819 23 1.714 2.069 2.500 2.807 24 1.711 2.064 2.492 2.797 25 1.708 2.060 2.485 2.787 26 1.706 2.056 2.479 2.779 27 1.703 2.052 2.473 2.771 23 1.701 2.048 2.467 2.763 29 1.699 2.045 2.462 2.756 30 1.697 2.042 2.457 2.750 Critical Regions] Rejection Regions Critical region, also known as the rejection region, describes the entire area of values that indicates you reject the null hypothesis. In other words, the critical region is the area encompassed by the values not included in the acceptance region. It is the area of the \"tails\" of the distribution. The \"tails\" of a test are the values outside of the critical values. In other words, the tails are the ends of the distribution and they begin at the greatest or least value in the alternative hypothesis (the critical values). Rejection Region If Population Variance Is Known To determine the critical region for a normal distribution, we use the table for the standard normal distribution. If the level of signicance is a = 0.10, then for a onetailed test, the critical region is below 2 = 1.28 or above 2 = 1.28. For a twotailed test, use %= 0.05 and the critical region is below z = 1.645 and above 2 : 1.645. If the absolute value of the calculated statistics has a value equal to or greater than the critical value, then the null hypotheses Ho should be rejected and the alternate hypothesis H 6118 assumed to be supported. Rejection Region If Population Variance Is Unknown To determine the critical region for a tdistribution, we use the table of the tdistribution. (Assume that we use a tdistribution with 20 degrees of freedom.) If the level of significance is a = .10, then for a onetailed test, t = 1.325 or t = 1.325. For a twotailed test, use %= 0.05 and then I: = 1.725 and t = 1.725. If the absolute value of the calculated statistics has a value equal to or greater than the critical value, then the null hypotheses Ho will be rejected and the alternate hypotheses Ha is assumed to be correct. Hypothesis Test and Their Tails There are three types of test from a \"tails\" standpoint: . A lefttailed test only has a tail on the left side of the graph. rejection region rejection region 43 .3 'I III 1 2 3 o A twotailed test has tails on both ends of the graph. This is a test where the null hypothesis is a claim of a specic value. rejection region zej eetian region Illustrative Examples: Determine the critical values and the appropriate rejection region. Sketch the sampling distribution. 1. Right-tailed test where a is known, 20.05, and n=34 In this example, the population standard deviation is known. Therefore, the test statistic would be ztest. To obtain the critical value for the level of significance of 0.05 and onetailed test, zvalue from the table is 1.645. The hypothesis test is righttailed, so the inequality symbol would be 2. Hence, the rejection region for a onetailed test is z 2 1.645. To sketch the graph, locate rst the critical value of 1.645 which is between the 1 and 2 in the normal curve. Then, shade the region greater than the critical value because it is a righttailed test. critical value rejection region 2:1.645 \"NA 3 2 '1 El 1 2 3 2. Two-tailed test where a is unknown, 20.05, and 11210 Since this is a twotailed test, 1/2 of 0.05= 0.025 of the values would be in the left and the other 0.025 would be in the right tail. Looking up tscore (n=101=9) associated with 0.025 on the reference table, we find 2.262. Therefore, +2.262 is the critical value of the right tail and 2.262 is the critical value of the left tail. The rejection region is 2.262 S t 2 2.262. critical value I = +1262 rejection region rcj action rcgic n 3. Left-tailed test where a" is known, 120.01, and n=40 A onetailed test with 0.01 would have 99% of the area under the curve outside of the critical region. Since the variance is known, we use zscore as the reference to find the critical value. This is a lefttailed test, so the critical value we need is negative. The solution is z= 2. 326. The rejection region is 2 S -2. 326. rejection ' , critical value 2 = 2.326 3 2 '1 El '1 2 3 4. A survey reports a customer in the drive thru lane of one fast food chain spends eight minutes to wait for his / her order. A sample of 24 customers at the drive thru lane showed mean of 7.5 minutes with a standard deviation of 3.2 minutes. Is the waiting time at the drive thru lane less than that of the survey made? Use 0.05 signicance level. Hypotheses Hypothesis Population Standard Known [Unknown a' is unknown. A onetailed test with 0. 05 level of significance has 95% of the area under the curve outside of the critical region. Since the variance is unknown, we use tscore with df = 241 =23 as the reference to determine the critical value. This is a lefttailed test, so the critical value we need is negative. The critical value is 2. 069 and the rejection region is tS2.069. rejectia I ragio critical value -_':"" r=2.m9 5. A banana company claims that the mean weight of their banana is 150 grams with a standard deviation of 18 grams. Data generated from a sample of 49 bananas randomly selected indicated a mean weight of 153.5 grams per banana. Is there sufcient evidence to reject the company's claim? Use 0:20.05. Hypotheses Hypothesis Population Level of Number z-value Test Standard Signicance of or Known / Unknown Sample t-value The rejection region is z S 1.96 or z 2 1.96. critical value critical value z:1.96 z=1.96 rejection regign refection region Jr -2 -1 0 l 2 3