Section 8.2 Day 4

Tomorrow: Quiz 8.2 --Use backside of note card used for 8.1 HW Quiz 8.1 – 8.2…when? Test 8.1 – 8.2…when? -- Both sides of 1 note card

Reject, not reject, or accept null hypothesis? P-value = 0.049 95% confidence 2) P-value = 0.049 99% confidence 3) P-value = 0.049 significance level of 0.1

Reject, not reject, or accept null hypothesis? P-value = 0.049 95% confidence P-value < α = 0.05 so reject Ho 2) P-value = 0.049 99% confidence 3) P-value = 0.049 significance level of 0.1

Reject, not reject, or accept null hypothesis? P-value = 0.049 95% confidence P-value < α = 0.05 so reject Ho 2) P-value = 0.049 99% confidence P-value > α = 0.01 so do not reject Ho 3) P-value = 0.049 significance level of 0.1

Reject, not reject, or accept null hypothesis? P-value = 0.049 95% confidence P-value < α = 0.05 so reject Ho 2) P-value = 0.049 99% confidence P-value > α = 0.01 so do not reject Ho 3) P-value = 0.049 significance level of 0.1 P-value < α = 0.1 so reject Ho

Types of Errors Reasoning of significance tests often compared to that of a jury trial.

Types of Errors Reasoning of significance tests often compared to that of a jury trial.

Types of Errors

Types of Error and Power Type I Error: rejecting a true null hypothesis

Types of Error and Power Type I Error: rejecting a true null hypothesis The probability of making a Type I error is equal to the significance level, α, of the test. P(Type I error) = α

Types of Error and Power Type I Error: To decrease the probability of a Type I error, make α smaller. We make α smaller by increasing the confidence level.

Types of Error and Power Type I Error: To decrease the probability of a Type I error, make α smaller. We make α smaller by increasing the confidence level. Confidence level α 90% 0.10 95% 0.05 99% 0.01

Types of Error and Power Type I Error: To decrease the probability of a Type I error, make α smaller. Changing sample size has no effect on the probability of a Type I error.

Types of Error and Power If the null hypothesis is false, you can not make a Type I error. Type I Error: rejecting a true null hypothesis

Types of Error and Power Type II Error: Failing to reject a false null hypothesis.

Types of Error and Power Type II Error: Failing to reject a false null hypothesis. To decrease the probability of making Type II error: Increase sample size or Make significance level, α, larger

Types of Error and Power If null hypothesis is true, you can not make a Type II error.

Types of Error and Power Power: the probability of rejecting the null hypothesis.

Types of Error and Power Power: the probability of rejecting the null hypothesis. When null hypothesis is false, you want to reject it. Therefore you want the power to be large.

Types of Error and Power When null hypothesis is false, you want to reject it. Therefore you want the power to be large. Power = 1 – probability of Type II error How do you increase the power?

Types of Error and Power Power = 1 – probability of Type II error How do you increase the power? decrease probability of Type II error

Types of Error and Power Power = 1 – probability of Type II error How do you increase the power? decrease probability of Type II error Take larger sample or make α larger

Page 511, P30

Page 511, P30 critical value associated with significance level of 0.12?

Page 511, P30 (a) critical value associated with significance level of 0.12? invNorm(0.06) is about ± 1.55

Page 511, P30 (b) What significance level is associated with critical values of z* of 1.73?

Page 511, P30 (b) What significance level is associated with critical values of z* of 1.73? 2[normalcdf(-1EE99, -1.73)] = 0.0836

Page 513, P42

Page 513, P42 (a) Name: One-sided significance test for a proportion One-sided because question asks whether poll results imply less than half of adults are satisfied.

Page 513, P42 Conditions: Problem states this is a random sample

Page 513, P42 Conditions: Problem states this is a random sample Both npo = 1000(0.5) = 500 and n(1 – po) = 1000(1 – 0.5) = 500 are at least 10.

Page 513, P42 Conditions: Problem states this is a random sample Both npo = 1000(0.5) = 500 and n(1 – po) = 1000(1 – 0.5) = 500 are at least 10. Number of adults in U.S. is at least 10(1000) = 10,000

Page 513, P42 (b) (Recall, null hypothesis has form p = po) H0 : p = 0.5, where p is proportion of all adults in the U.S. who would say they are satisfied with the quality of K–12 education in the nation

Page 513, P42 (b) (Recall, null hypothesis has form p = po) H0 : p = 0.5, where p is proportion of all adults in the U.S. who would say they are satisfied with the quality of K–12 education in the country Ha: p < 0.5 Question asks whether poll results imply less than half of adults are satisfied.

Page 513, P42 (c) po = 0.5, x = 46%(1000) = 460, n = 1000, p < po Use 1-PropZTest z is approx. -2.53; P-value is approx. 0.0057

Page 513, P42 (d) No significance level was given, so use α = 0.05 and confidence level of 95%. I reject the null hypothesis because the P-value of 0.0057 is less than the significance level of α = 0.05.

Page 513, P42 (d) No significance level was given, so use α = 0.05 and confidence level of 95%. I reject the null hypothesis because the P-value of 0.0057 is less than the significance level of α = 0.05. There is sufficient evidence to support the claim that less than a majority of adults in the United States are satisfied with the quality of K–12 education.

Page 513, P42 (d) I reject the null hypothesis because the test statistic, z, of about -2.53 is outside the critical value of -1.645. There is sufficient evidence to support the claim that less than a majority of adults in the United States are satisfied with the quality of K–12 education.

Page 511, P33 Name of test?

Page 511, P33 Two-sided significance test for a proportion. Carry out the four steps in a test of the null hypothesis that half of the bookstores in the United States sell DVDs. Hypotheses in symbols?

Page 511, P33 Two-sided significance test for a proportion. Carry out the four steps in a test of the null hypothesis that half of the bookstores in the United States sell DVDs. Ho: p = 0.5

Page 511, P33 Two-sided significance test for a proportion. Carry out the four steps in a test of the null hypothesis that half of the bookstores in the United States sell DVDs. Ho: p = 0.5 Ha: p ≠ 0.5

Page 516, E41 Name of test?

Page 516, E41 One-sided significance test for a proportion. Determine whether the increase in the proportion of mutations is statistically significant Hypotheses in symbols?

Page 516, E41 One-sided significance test for a proportion. Determine whether the increase in the proportion of mutations is statistically significant Ho: p = 0.02

Page 516, E41 One-sided significance test for a proportion. Determine whether the increase in the proportion of mutations is statistically significant Ho: p = 0.02 Ha: p > 0.02

Page 514 E27 Name of test?

Page 514 E27 Two-sided significance test for a proportion test the claim that 60% of the students in school carry backpacks to school

Page 515, E33

Page 515, E33

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis The hypothesis you assume is true in the test

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis The hypothesis you assume is true in the test

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis Rejecting Ho when Ho is really true

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis Rejecting Ho when Ho is really true

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis z-score of the sample statistic

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis z-score of the sample statistic

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis Half the width of the confidence interval

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis Half the width of the confidence interval

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis Not rejecting Ho when Ho is really false

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis Not rejecting Ho when Ho is really false

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis The hypothesis you are seeking evidence for

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis The hypothesis you are seeking evidence for

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis The probability of seeing a result this extreme or more extreme

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis The probability of seeing a result this extreme or more extreme

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis statistic +/- (critical value of z)(std dev of statistic)

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis statistic +/- (critical value of z)(std dev of statistic)

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis A standard for Ho rejection: also called z*

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis A standard for Ho rejection: also called z*

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis A standard for Ho rejection: also called the alpha-level

Vocabulary A) margin of error B) z-statistic C) Type I error D) Type II error E) critical value of z F) significance level G) P-value H) confidence interval I) null hypothesis J) alternative hypothesis A standard for Ho rejection: also called the alpha-level

Page 511, P33

Page 511, P33 Step 1: Two-sided significance test for a proportion.

Page 511, P33 Conditions: (1) Problem states this was random sample

Page 511, P33 Conditions: (1) Problem states this was random sample (2) npo = 500(0.5) = 250 and n(1 – po) = 500(0.5) = 250 are both at least 10

Page 511, P33 Conditions: (1) Problem states this was random sample (2) npo = 500(0.5) = 250 and n(1 – po) = 500(0.5) = 250 are both at least 10 (3) Total number of bookstores in U.S. is greater than 10(500) = 5000.

Page 511, P33 Step 2: State hypotheses. Ho: p = 0.5, where p is proportion of all U.S. bookstores that sell DVDs

Page 511, P33 Step 2: State hypotheses. Ho: p = 0.5, where p is proportion of all U.S. bookstores that sell DVDs Ha: p ≠ 0.5

Page 511, P33 Step 3: Compute test statistic and draw sketch

Page 511, P33 Step 3: Compute test statistic and draw sketch 1-PropZTest po: 0.5 x: 265 n: 500 prop ≠ po

Page 511, P33 Step 3: Compute test statistic and draw sketch 1-PropZTest po: 0.5 x: 265 z ≈ 1.34 n: 500 P-value ≈ 0.18 prop ≠ po

Page 511, P33 Not given level of significance so use  = 0.05 and critical values of z* = ± 1.96 Step 3 (con’t):

Page 511, P33 Step 4: Write conclusion in context.

Page 511, P33 Step 4: Write conclusion in context. I do not reject the null hypothesis because the P-value of 0.18 is greater than the significance level of a = 0.05.

Page 511, P33 Step 4: Write conclusion in context. I do not reject the null hypothesis because the P-value of 0.18 is greater than the significance level of a = 0.05. There is not sufficient evidence to support the claim that the proportion of bookstores in the U.S. that sell DVDs is not half.

Page 513, P42 Hypotheses?

Page 513, P42 Ho: p = 0.5, where p is the proportion of adults in the U.S. who say they are satisfied with the quality of K-12 education in the nation.

Page 513, P42 Ho: p = 0.5, where p is the proportion of adults in the U.S. who say they are satisfied with the quality of K-12 education in the nation. Ha: p < 0.5 Does this imply that less than a majority are satisfied?

Page 512, P35

Page 512, P35 B. The proportion of all households that are multigenerational this year.

Page 513, E25 Which of these is not a true statement?

Page 513, E25 Which of these is not a true statement? (The 95% confidence interval is 0.82 to 0.96). D. If 75% of all dogs wear a collar, then you are reasonably likely to get a result like the one from this sample.

Page 515, E35

Page 515, E35 Suppose the newspaper’s percentage is actually right.

Page 515, E35

Page 515, E36

Page 515, E36 Suppose the newspaper’s percentage is actually wrong.

Page 515, E36 Suppose the newspaper’s percentage is actually wrong.

Page 516, E41

Page 516, E41

Page 516, E41

Page 516, E41

Page 516, E41

Page 511, P28

Page 511, P28 B. Assuming that Ho is true, the P-value is the probability of observing a value of a test statistic at least as far out in the tails of the sampling distribution as is the value of the test statistic, z, from your sample.

Questions?

Suppose that in a random sample of 500 households, you find that 309 households have a computer. Test the claim that 56.5% of all households in the United States have a computer. Use a significance level of α = 0.01.

Name of Test This is a two-sided significance test for a proportion.

Conditions 1) We are told this is a random sample from a binomial population.

Conditions 1) We are told this is a random sample from a binomial population. 2) npo = 500(0.565) = 282.5 n(1 – po) = 500(0.435) = 217.5; so both are at least 10

Conditions 1) We are told this is a random sample from a binomial population. 2) npo = 500(0.565) = 282.5 n(1 – po) = 500(0.435) = 217.5; so both are at least 10 3) the number of households in the U.S. is at least 10(500) = 5000

Hypotheses Ho: p = 0.565, where p is the proportion of households in the U.S. that have a computer

Hypotheses Ho: p = 0.565, where p is the proportion of households in the U.S. that have a computer Ha: p

Suppose that in a random sample of 500 households, you find that 309 households have a computer. Test the claim that 56.5% of all households in the United States have a computer. Use a significance level of α = 0.01.

Test Statistic and P-value 1-PropZTest po: 0.565 x: 309 n: 500 prop po

Test Statistic and P-value

Conclusion Do not reject the null hypothesis at the 0.01 significance level because the P-value of 0.0168 is greater than 0.01. There is not sufficient evidence to support the claim that the percent of all households in the U.S. that have computers is not 56.5%.

Conclusion Do not reject the null hypothesis at the 99% confidence level because the test statistic of ± 2.39 is within the critical values interval of ± 2.576. There is not sufficient evidence to support the claim that the percent of all households in the U.S. that have computers is not 56.5%.