5/18/2018 AP Stats CIs for Means Homework 13 1. Name: 1 Proportion Z-Test Conditions and Assumptions: Stated it is a simple random sample np0 = 15 > 10 and n(1- p0) = 15 > 10 The population of students at THS is greater than 10 times the sample size (300) Hypotheses: st H 0 :The true population proportion of THS students with a 1 quarter GPA of an A or B is 50%. st H A :The true population proportion of THS students with a 1 quarter GPA of an A or B is greater than 50%. pˆ p z 0 Formula: p0 (1 p0 ) n 19 / 30 0.5 z Calculations: 0.5(1 0.5) = 1.460593487 30 p-value = 0.072063555 Sketch:
Conclusion: Based upon a p-value of 0.072063555, there is sufficient evidence to reject the null hypothesis that the true population proportion of THS students with a 1st quarter GPA of an A or B is 50% at the 0.10 level. If the null hypothesis were true, the probability of getting a pˆ of 19/30 or more extreme is 0.072063555. 2. No, the null hypothesis we correctly rejected. **If you used a = 0.05level and failed to rejected, then a type II error was made. 3. Name: 1 Proportion Z-Interval Conditions and Assumptions: Stated it is a simple random sample n pˆ = 19 > 10 and n(1- pˆ ) = 11 > 10 The population of students at THS is greater than 10 times the sample size (300) pˆ(1 pˆ) Formula: pˆ z * n 5/18/2018
0.6333(1 0.63333) Calculations: 0.6333 1.43953 = 0.50668 to 0.75999 30 Conclusion: We are 85% confident that the true population proportion of THS students in the 1st quarter with an A or B average is between 0.50668 and 0.75999. If we constructed 100 confidence intervals of size 30, we would expect 85 of these to capture the true population proportion. 4. Name: t-test Conditions and Assumptions: Stated it is a simple random sample The sample size of 50 is sufficiently large (> 40). Hypotheses: st H 0 :The true population mean GPA of THS students in the 1 quarter is 3.2. st H A : The true population mean GPA of THS students in the 1 quarter is less than 3.2. x t 0 Formula: s n 3.0615- 3.2 t = Calculations: 0.910199 = -1.075965687 50 Degrees of Freedom = 49 p-value = 0.1436059773 Sketch:
Conclusion: Based upon a p-value of 0.1436059773, there is NOT sufficient evidence to reject the null hypothesis that the true population mean GPA of THS students in the 1st quarter is 3.2 at any reasonable significance level. If the null hypothesis were true, the probability of getting a x of 3.2 or more extreme is 0.1436059773. 5. Yes, Type II Error the null hypothesis should have been rejected and it was not. 6. Name: t-Interval Conditions and Assumptions: Stated it is a simple random sample The sample size of 50 is sufficiently large (> 40). 5/18/2018
s Formula: x t * n 0.910199 Calculations: 3.0615 1.676550835 = 2.8457 to 3.2773 50 Degrees of Freedom = 49 Conclusion: We are 90% confident that the true population mean GPA of THS students in the 1st quarter is between 2.8457 and 3.2773. If we constructed 100 confidence intervals of size 50, we would expect 90 of these to capture the true population proportion. 7. Name: Chi Square Goodness of Fit Test Conditions and Assumptions: Expected Counts are A – 30, B – 25, C – 25, D – 15, and F – 5 which are all 5 or bigger. States it is a random sample. Hypotheses: st H 0 :The true distribution of 1 quarter grades for THS is A – 30%, B – 25%, C – 25%, D – 15%, and F – 5%.
H A : At least one of the proportions is not as advertised. (O E)2 Formula: 2 E Calculations (29 30) 2 (32 25) 2 (29 250) 2 (7 15) 2 (3 5)2 = 7.7 30 25 25 15 5
Degrees of Freedom = 4
p-value = 0.1032067217 Sketch
Conclusion Based upon the p-value of 0.1032067217, there is not sufficient evidence to reject the null hypothesis that the distribution of the digits of grade for the 1st quarter at THS is as claimed at any reasonable significance level. If the null hypothesis 5/18/2018
were true the probability of getting a test statistic of 7.70 or more extreme is 0.1032067217. 8. Yes, Type II Error the null hypothesis should have been rejected and it was not. 9. 1797 10. 0.7127880455 11. 64 12. 337