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Confidence interval for the population mean - μ: ci = X tcv (SX) or ci = X tcv (σX) S Sx Note: SX is the standard error of the mean. n Or x n When the population standard deviation (σ) is known df = n-1 Example: Find the 99% confidence interval for : Data: N = 25, X = 75, S = 15 15 Sx 25 3.00 df = 24 tcv = 2.797 (See table B) ci = 75 2.797 (3) = 75 8.391 = 66.61 to 83.39 Margin of error for a proportion (ME) Critical Values Z CI 1.00 68% 1.65 90% 1.96 95% 2.58 99% MEci = Sp x Zci Sp pq n Note: Sp is the Standard Error of the Proportion Example: find the ME ci=95% Data: n = 100 p = .25, q = .75 Sp .25 x .75 .043 100 Zci = 1.96 MEci = .043 x 1.96 = .085 (or 8.5%) 1 T-Test for a Single Sample Mean (X ) verses a Known Population Mean (µ). X - µ Sx t= Note: df = n-1 Note: Sx = S/√n The computed t is compared to the tcv (critical value of t See Table at end of document), for a given df , -level, and directional v non-directional alternative hypothesis). If the computed t reaches or exceeds the tcv then the Ho is rejected. Null Hypothesis: Ho X = µ (Single population) 1. 2. 3. 4. The sample mean (X ) does not differ significantly from the known population mean (µ). Any difference between the sample mean (X ) and the population mean (µ) can be attributed to random sampling error. The sample comes from a population with a mean equal to µ . The research hypothesis is not supported by the data. Alternative Hypotheses (choose only one) from Ha1, Ha2, or Ha3 below: X Non-directional, 2-tail test, ±tcv 1 ≠ µ2) The sample mean (M) differs significantly from the known population mean (µ). The difference between the sample mean (X ) and the population mean (µ) cannot be attributed to random sampling error. 3 The sample does not come from a population with a mean is equal to µ. 4 The null hypothesis is not supported by the data. Ha2 X > µ (Two populations: µ1> µ2) Directional, 1-tail test, +tcv 1 The sample mean (X) is significantly greater than the known population mean (µ). 2 The difference between the sample mean (X ) and the population mean (µ) cannot be attributed to random sampling error. 3 The sample comes from a population with mean with mean that is greater than µ . 4 The null hypothesis is not supported by the data, the research hypothesis is. Ha3 X < µ (Two populations: µ1< µ2) Directional, 1-tail test, -tcv 1 The sample mean (M) is significantly less than the known population mean (µ). 2 The difference between the sample mean (M) and the population mean (µ) cannot be attributed to random sampling error. 3 The sample comes from a population with mean that is less than µ . 4 The null hypothesis is not supported by the data, the research hypothesis is. Example Data: X = 75, N = 25, S = 15, = 100. Test to see if the sample mean (75) differs significantly from the population mean (100). Set alpha (’’) = .01. Ha1 1 2 df = (25-1) = 24 tcv = 2.797 Ho Ha1 X = µ Sx = 15 /25 = 3.00 t= 75-100 Because the computed t (- 8.33) exceeds tcv ( 2.797), the Null Hypothesis (Ho) ---------- = 8.33 is rejected , and the Alternative Hypothesis (Ha1) is supported. 3.00 The sample mean is significantly below the population mean. 2 t-test for Independent Groups: Note: Sx-x = t SS1 SS 2 1 1 n1 n2 2 n1 n2 X1 X 2 SX1 SX 2 Note: df = n1+n2-2 Note: SX -X is the Standard Error of the difference between means – Independent Samples The computed t is compared to the tcv (critical value of t from at end of this document), for a given df , -level, and directional v non-directional alternative hypothesis). If the computed t reaches or exceeds the tcv then the Ho is rejected. Null Hypothesis Ho M1 = M2 (Single population) 1 Sample Mean-1 (M1 ) does not differ significantly from the Sample Mean-2 (M2 ). 2 Any difference between the two sample means can be attributed to random sampling error 3 The samples come from the same population 4 The research hypothesis is not supported by the data. Alternative Hypotheses (choose only one) from Ha1, Ha2, or Ha3 below: Ha1 M1 M2 (Two populations: µ1 ≠ µ2) Non-directional, 2-tail test, ±tcv 1 Sample Mean-1 (M1 ) differs significantly from Sample Mean-2 (M2 ). 2 The difference between the sample means cannot be attributed to random sampling error. 3 The samples do not come from a single (the same) population. 4 The null hypothesis is not supported by the data. Ha2 M1 > M2 (Two populations: µ1 > µ2) Directional, 1-tail test, +tcv 1 Sample Mean-1 (M1 ) is significantly greater than Sample Mean-2 (M2 ). 2 The difference between the sample means cannot be attributed to random sampling error. 3 Sample 1 comes from a population with a mean that is greater than Sample -2. 4 The null hypothesis is not supported by the data, the research hypothesis is. Ha3 M1 < M2 (Two populations: µ1 < µ2) Directional, 1-tail test, -tcv 1 Sample Mean-1 (M1 ) is significantly less than Sample Mean-2 (M2 ). 2 The difference between the sample means cannot be attributed to random sampling error. 3 Sample -1 comes from a population with a mean that is less than Sample -2 . 4 The null hypothesis is not supported by the data, the research hypothesis is. Example: Test to see if males and females differ significantly on the General Creativity Test. Set ’’ = .05. Group 1 Group 2 Males Females Means 100 107 SS 1000.825 1500.02 N 21 31 df = (21+31-2) = 50 Ho M males = M females tcv = 2.0086 Ha1 M males ≠ M females Sx-x = 1000.825 1500.02 1 1 21 31 2 21 31 = 2.00 100 – 107 = -3.5 2.00 Because the computed t (- 3.5) exceeds the tcv ( 2.0086), the Null Hypothesis (Ho) is rejected and the Alternative Hypothesis (Ha1) is supported. The two sample means differ significantly. On average males score significantly lower than females on the General Creativity Test. t= 3 t-test for dependent/Correlated Groups: t D D D 2 - (D) 2 /n n(n 1) D n Note df = n cases -1 The computed t is compared to the tcv (critical value of t Table at the end of this document) for a given df , -level, and directional v non-directional alternative hypothesis). If the computed t reaches or exceeds the tcv then the Ho is rejected. 1 2 3 Null Hypothesis: Ho D0 Time-1 mean (M time-1 ) does not differ significantly from Time-2 mean (M time-2 ). Any difference between the two means can be attributed to random sampling error. The research hypothesis is not supported by the data. Alternative Hypotheses (choose only one) from Ha1, Ha2, or Ha3 below: Ha1 D 0 (Mtime-1 Mtime-2) Non-directional, 2-tail test, ± tcv 1 2 3 Ha2 Mean-1 (Mtime-1 ) differs significantly from Mean-2 (Mtime-2 ). The difference between the sample means cannot be attributed to random sampling error. The null hypothesis is not supported by the data. D 0 (M time-1 > M time-2) Directional, 1-tail test, + tcv 1 2 3 Ha3 Time-1 mean (Mtime-1 ) is significantly greater than Time-2 mean (Mtime-2 ). The difference between the means cannot be attributed to random sampling error. The null hypothesis is not supported by the data, the research hypothesis is. D 0 (Mtime-1 < M time-2) Directional, 1-tail test, -tcv 1 2 3 Time-1 mean (Mtime-1 ) is significantly less than Time-2 mean (Mtime-2 ). The difference between the means cannot be attributed to random sampling error. The null hypothesis is not supported by the data, the research hypothesis is. Example: Test to see if the mean SAT score for the sample of 5 students changes significantly following an SAT review course. Set ’’ = .05. Ho D 0 (M time-1 = Mtime-2) df = 4 Ha D 0 (M time-1 Mtime-2) SAT-Pretest SAT-Posttest D = -150 t Mean 1120 1150 2 D = 20,300 30 20,300 - (-150) 2 /5 5(4 1) tcv = 2.7764 S 286.36 326.27 N = 5 D = -30 1.07 Because the computed t (- 1.07) does not reach or exceed the tcv ( 2.7764) the Null Hypothesis (Ho) is accepted. The mean pretest score does not differ significantly from the mean posttest-test score. On average student performance on the SAT does not change significantly following the Review Course. The SAT Review Course does not significantly improve SAT performance. 4 Z-test for a single sample proportion (p) verses a known population proportion (P) The computed Z is compared to the Zcv for a given -level, and directional or non-directional alternative hypothesis. If the computed Z reaches or exceeds the Zcv then the Ho is rejected. Alpha Level (α) α’’ Two Tail Values of Z α’ One Tail Values of Z α. = 05 (5% significance level) Z = ± 1.96 Z = 1.65 α. = 01 (1% significance level) Z = ± 2.58 Z = 2.33 Null Hypothesis: Ho p=P 4. The sample proportion (p) does not differ significantly from the known population proportion (P). 5. Any difference between the sample proportion (p) and the population proportion (P) can be attributed to random sampling error. 6. The sample comes from a population with a proportion equal to P. 7. The research hypothesis is not supported by the data. Alternative Hypotheses (choose only one) from Ha1, Ha2, or Ha3 below: p (Two populations: P1 ≠ P2) Non-directional, 2-tail test, ±Zcv The sample proportion (p) differs significantly from the known population proportion (P). The difference between the sample proportion (p) and the population proportion (P) cannot be attributed to random sampling error. 3 The sample does not come from a population with a proportion is equal to P. 4 The null hypothesis is not supported by the data. Ha2 p > P (Two populations: P1> P2) Directional, 1-tail test, +Zcv 1 The sample proportion (p) is significantly greater than the known population proportion (P). 2 The difference between the sample proportion (p) and the population proportion (P) cannot be attributed to random sampling error. 3 The sample comes from a population with proportion (p) that is greater than P. 4 The null hypothesis is not supported by the data, the research hypothesis is. Ha3 p < P (Two populations: P1< P2) Directional, 1-tail test, -Zcv 1 The sample proportion (p) is significantly less than the known population proportion (P). 2 The difference between the sample proportion (p) and the population proportion (P) cannot be attributed to random sampling error. 3 The sample comes from a population with proportion that is less than P. 4 The null hypothesis is not supported by the data, the research hypothesis is. Problem 1 (Z-test for a sample proportion (p) v. a known population proportion (P): Dr. Spock, the Medical Director of Snob Hill Hospital, boasts that Snob Hill’s community outreach program for expectant mothers has resulted in significantly healthier babies. To assess his claim, he randomly birth records for 200 recent births at Snob Hill. Among the 200 births, 16 babies were born premature. Test to see if the rate of premature births at Snob Hill Hospital is significantly different than the National rate of 12% ( = .12). Set = .05. Ha1 1 2 1 2 3 4 5 Z State the Null and Alternative Hypothesis in symbols and words. Zcv Z= Do you reject or accept the Ho? Discuss the goals of the study, the results, conclusions and recommendations that can be drawn from the study. pP Sp Note: Sp pq N Sp is the standard error of the proportion 5 Z-test for two independent samples (p1 v p2). The computed Z is compared to the Zcv for a given -level, and directional, or non-directional alternative hypothesis. If the computed Z reaches or exceeds the Zcv then the Ho is rejected. Null Hypothesis Ho p1 = p2 (Single population) 1 Sample Proportion-1 (p1) does not differ significantly from the Sample Proportion-2 (p2). 2 Any difference between the two samples Proportions can be attributed to random sampling error 3 The samples come from the same population 4 The research hypothesis is not supported by the data. Alternative Hypotheses (choose only one) from Ha1, Ha2, or Ha3 below: Ha1 p1 p2 (Two populations: P1 ≠ P2) Non-directional, 2-tail test, ±Zcv 1 Sample Proportion-1 (p1) differs significantly from Sample Proportion-2 (p2). 2 The difference between the samples Proportions cannot be attributed to random sampling error. 3 The samples do not come from a single (the same) population. 4 The null hypothesis is not supported by the data. Ha2 1 2 3 4 p1 < p2 (Two populations: P1 > P2) Directional, 1-tail test, +Zcv Sample proportion-1 (p1) is significantly greater than Sample proportion-2 (p2). The difference between the samples proportions cannot be attributed to random sampling error. Sample 1 comes from a population with a proportion that is greater than the p observed in Sample -2. The null hypothesis is not supported by the data, the research hypothesis is. Ha3 1 2 3 4 p1 < p2 (Two populations: P1 < P2) Directional, 1-tail test, -Zcv Sample Proportion-1 (p1) is significantly less than Sample Proportion-2 (p2). The difference between the samples Proportions cannot be attributed to random sampling error. Sample -1 comes from a population with a proportion that is less than the p observed in Sample -2. The null hypothesis is not supported by the data, the research hypothesis is. Problem 2 (Z test for two sample proportions) : Dr. Beeper, the Medical Director Bunker Hill Hospital (BHH), claims that the Bunker Hill Hospital’s community outreach program for expecting mothers is superior to the Snob Hill Hospital’s program (see problem 1, above, where p = .08, and n = 200). Based on recent birth records, Dr. Beeper reports that among 300 BHH births, 30 were premature births. Test to see if the rate of premature births at BHH differs significantly from the rate at Snob Hill Hospital. Set = .05. 1 2 3 4 5 State the Null and Alternative Hypothesis in symbols and words. Zcv Z= Do you reject or accept the Ho? Discuss the goals of the study, the results, conclusions and recommendations that can be drawn from the study. Z= p1 - p2 Sp-p Note Sp-p = p1q1 p 2 q2 n1 n2 Sp-p is the standard error of the difference between proportions 6
