Statistics Final Exam

Statement for a confidence interval

"I am C% confident that the interval from X% to Y% captures the true percentage of who/what that do this/that"

A medical researcher measured the pulse rates of a sample of randomly selected adults and found the following students t-based confidence interval: 95% confidence, 70.887604 < mu(pulse) < 74.497011 Explain carefully what the software output means.

Based on this sample, we can say with 95% confidence that the mean pulse rate of adults is between 70.9 and 74.5 bpm

After getting trounced by your little brother in a children's game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? State your conclusion.

Because the P-value is high, do not reject Ho. The data show no evidence that the die is unfair.

In 2015, the U.S. Census Bureau reported that 62.2% of American families owned their homes- the lowest rate in 20 years. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers to encourage people to become home owners. They decide to adopt a plan on a 2-year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. What would the power of the test represent in this context?

City's ability to detect actual increase in home ownership

If p-value is larger than alpha

Failure to reject null hypothesis. There is not enough evidence to suggest Ha.

Is a coin fair? Write a null and alternative hypothesis.

H0: p= 0.50 Ha: p is not equal to 0.50

After getting trounced by your little brother in a children's game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? State your hypothesis.

Ho: the die is fair Ha: the die is not fair

conditions and assumptions for one-mean t interval

Independence Assumption (RC) and Normal Population Assumption (roughly unimodal and symmetric)

Conditions for a mean hypothesis test

RC, NNC

What is the shape in a chi-square model?

Skewed right

H0: Not guilty, Ha: guilty What's the Type I error?

When we come to the conclusion that they are guilty but are actually innocent

H0: Not guilty, Ha: guilty What's the Type II error?

When we come to the conclusion that they are innocent but were actually guilty.

Formula for z score

observation-mean/standard deviation

If p-value is smaller than alpha

reject the null hypothesis. There is evidence to suggest Ha.

power

the probability that we correctly reject a false null hypothesis

Type I error (alpha error)

when a researcher rejects the null hypothesis when it is actually true

A poll conducted by the University of Montana classified respondents by whether they were male or female and political party. We wonder if there is evidence of an association between being male or female and party affiliation. Find the p-value for your test.

x^2= 4.851 df2. Pvalue= 0.0084

After getting trounced by your little brother in a children's game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? Find x^2 and the P-value.

x^2= 5.600 P-value= 0.3471

If your proportion interval is ( - , + ), what does this mean?

zero is inside the interval so there is no difference between groups

After getting trounced by your little brother in a children's game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? How many degrees of freedom are there?

5

goodness of fit test

A test of whether the distribution of counts in one categorical variable matches the distribution predicted by a model

A poll conducted by the University of Montana classified respondents by whether they were male or female and political party. We wonder if there is evidence of an association between being male or female and party affiliation. Are the conditions satisfied?

Counted data are counts for political party and sex, probably independent, all expected frequencies are greater than 5.

T/F: A high p-value shows that the null hypothesis is true.

False, a high P-value shows data consistent with the null hypothesis but does not necessarily make it true.

T/F: A P-value below 0.05 is always considered sufficient evidence to reject a null hypothesis.

False, depends on the limit you set for alpha.

T/F: A very low P-value proves that the null hypothesis is false.

False, it results in rejection of the null hypothesis but cannot necessarily prove it false.

Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Is this statement true? For a given confidence level, halving the margin of error requires a sample twice as large.

False.

Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Is this statement true? For a given sample size, higher confidence means smaller margin of error.

False.

T/F: A very high P-value is strong evidence that the null hypothesis is false.

False. A very high P-value shows data that is consistent with the null hypothesis, but does not necessarily make it true.

Only about 20% of people who try to quit smoking succeed. Sellers of a motivational tape claim that listening to the recorded messages can help people quit. Write a null and alternative hypothesis.

H0: p = 0.20 Ha: p > 0.20

In 2015, the U.S. Census Bureau reported that 62.2% of American families owned their homes- the lowest rate in 20 years. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers to encourage people to become home owners. They decide to adopt a plan on a 2-year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. In words, what will their hypothesis be?

H0: p= home ownership will not increase Ha: p= home ownership will increase

A poll conducted by the University of Montana classified respondents by whether they were male or female and political party. We wonder if there is evidence of an association between being male or female and party affiliation. Write an appropriate hypothesis.

Ho: Sex and political party are independent. Ha: There is a relationship between sex and political party.

Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the mean weight gain this supplement produces has a margin of error of +/- 11 pounds. Some students wrote the following conclusions. Are they correct? Explain. If this supplement is tested on another sample of cows, there is a 95% chance that their average weight gain will be between 45 and 67 pounds.

No. This interval is not a standard. There is a 95% chance that another sample will have its average weight gain within two standard errors of the mean.

A company's old antacid formula provided relief for 70% of the people who used it. The company tests a new formula to see if it is better and gets a p-value of 0.27. Is it reasonable to conclude that the new formula and the old one are equally effective? Explain.

No. We can say only that there is a 27% chance of seeing the observed effectiveness just from natural sampling variation. There is insufficient evidence that the new formula is more effective, but we can't conclude that they are equally effective.

The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 34% of students had not been absent from school even once during the previous month. In a 2000 survey, responses from 8302 students showed that this figure had slipped to 33%. Officials would, of course, be concerned if student attendance were declining. Do these figures give evidence of a change in student attendance? Perform the test and find the P-value.

P= 0.054 or 0.055 using tables (????)

Type II error (beta error)

The probability of failing to reject a null hypothesis that is, in fact, false

If your proportion interval is ( +, + ), what does this mean?

There will be a difference between groups, the first group experienced larger values.

If your proportion interval is ( - , - ), what does this mean?

There will be a difference between groups, the second group observed larger values

assumptions and conditions for comparing counts

counted data condition (counts for a categorical variable), independence assumption, expected cell frequency condition (see at least 5 individuals in each cell of contingency table)

After getting trounced by your little brother in a children's game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? To see if these results are unusual, will you test goodness of fit, homogeneity, or independence?

goodness of fit

conditions and assumptions for comparing group means

independence assumption (RC), independent groups assumption, nearly normal condition

conditions and assumptions for paired sample tests

paired data condition (data must be paired), Independence assumption, nearly normal condition

Researchers at the national cancer institute released the results of a study that investigated the effect of weed-killing herbicides on house pets. They examined 827 dogs from homes where an herbicide was used on a regular basis, diagnosing malignant lymphoma in 473 of them. Of the 130 dogs from homes where no herbicides were used, only 19 were found to have lymphoma. Construct a 95% confidence interval for this difference.

(0.356, 0.495)

It is believe that as many as 25% of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group. What sample size would produce a 3% margin of error?

(p-hat x q-hat x Z*90^2) / n^2 = .25 x .75 x 1.645^2 / 0.03^2 = 564

It is believe that as many as 25% of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group. Suppose we want to cut the margin of error to 4%. What's the necessary sample size?

(p-hat x q-hat x Z*90^2) / n^2 = .25 x .75 x 1.645^2 / 0.04^2 =318

It is believe that as many as 25% of adults over 50 never graduated from high school. We wish to see if this percentage is the same among the 25 to 30 age group. How many of this younger age group must we survey in order to estimate the proportion of non-grads to within 6% with 90% confidence.

(p-hat x q-hat x Z*90^2) / n^2 = .25 x .75 x 1.645^2 / 0.06^2 = 141

Researchers at the national cancer institute released the results of a study that investigated the effect of weed-killing herbicides on house pets. They examined 827 dogs from homes where an herbicide was used on a regular basis, diagnosing malignant lymphoma in 473 of them. Of the 130 dogs from homes where no herbicides were used, only 19 were found to have lymphoma. What's the standard error of the difference in two proportions?

0.035

5 steps of a hypothesis test

1. Identify null and alternative hypothesis. 2. Verify necessary conditions/assumptions 3. Draw a picture. 4. Perform all necessary computations 5. Write a conclusion stating p-value and what it means.

Conditions that need to be met for sampling distribution models and confidence intervals for PROPORTIONS

1. Independence assumption 2. Randomization condition 3. 10% condition 4. Success/Failure condition

Conditions that need to be met for confidence intervals for MEANS

1. Independence assumption (Randomization condition) 2. Normal Population Condition (Nearly Normal Condition: the data come from a unimodal and symmetric distribution)

properties of correlation coefficient

1. the sign of a correlation coefficient gives the direction of the association 2. correlation is always between -1 and +1 (inclusive). 3. Correlation treats x and y symmetrically. 4. Correlation has no units. 5. Correlation is not affected by changes in the center or scale of either variable. 6. Correlation measures the strength of the linear association between two variables. 7. Correlation is sensitive to outliers.

A medical researcher measured the pulse rates of a sample of randomly selected adults and found the following students t-based confidence interval: 95% confidence, 70.887604 < mu(pulse) < 74.497011 What's the margin of error for this interval?

1.8bpm

After getting trounced by your little brother in a children's game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? If the die is fair, how many times would you expect each face to show?

10

The Core Plus Mathematics Project (CPMP) is an innovative approach to teaching Mathematics that engages students in group investigations and mathematical modeling. After field tests in 36 high schools over a three-year period, researchers compared the performances of CPMP students with those taught using a traditional curriculum. In one test, students had to solve applied algebra problems using calculators. Scores for 320 CPMP students were compared to those of a control group of 273 students. Computer software was used to create a confidence interval for the difference in mean scores. Confidence level: 95% Variable: Mu(CPMP)-Mu(Control) Interval: (5.573, 11.427) What's the margin of error for this confidence interval?

2.927

A researcher developing scanners to search for hidden weapons at airports has concluded that a new device is significantly better than the current scanner. He made this decision based on a test using alpha=0.05. Would he have made the same decision at alpha=0.10 and alpha=0.01?

At alpha=.10 yes because if it passed at 0.05 it will pass at .10 since it is larger. We are unable to determine if it will pass at .01.

The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 34% of students had not been absent from school even once during the previous month. In a 2000 survey, responses from 8302 students showed that this figure had slipped to 33%. Officials would, of course, be concerned if student attendance were declining. Do these figures give evidence of a change in student attendance? Write an appropriate hypothesis.

H0: P2000 = 0.34 Ha: P2000 is not equal to 0.34.

A governor is concerned about his ''negatives''- the percentage of state residents who express disapproval of his job performance. His political committee pays for a series of TV ads, hoping that they can keep the negatives below 30%. They will follow up with a poll to assess ad effectiveness. Write a null and alternative hypothesis.

H0: p= 0.30 Ha: p < 0.30

Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated in England during the 1950s, but now there is concern that a generation of children more likely to watch TV or play computer games than spend time outdoors is at increased risk. A recent study of 2700 children randomly selected from all parts of England found 20% of them deficient in Vitamin D. What does your interval mean?

I am 98% confident that the interval from 18.2% to 21.8% captures the true percentage of English children who are Vitamin D deficient.

Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated in England during the 1950s, but now there is concern that a generation of children more likely to watch TV or play computer games than spend time outdoors is at increased risk. A recent study of 2700 children randomly selected from all parts of England found 20% of them deficient in Vitamin D. What does the "98% confidence" mean?

If we repeat this process over and over again with a sample size of 2700, we expect that 98% of all intervals to capture the true population parameter (English children deficient in Vitamin D).

conditions and assumptions for comparing group proportions

Independence assumption (RC), Independent Group Assumption (groups are independent of each other), 10% condition, S/F condition

A poll conducted by the University of Montana classified respondents by whether they were male or female and political party. We wonder if there is evidence of an association between being male or female and party affiliation. Is this a test of homogeneity or independence?

Independence.

The Core Plus Mathematics Project (CPMP) is an innovative approach to teaching Mathematics that engages students in group investigations and mathematical modeling. After field tests in 36 high schools over a three-year period, researchers compared the performances of CPMP students with those taught using a traditional curriculum. In one test, students had to solve applied algebra problems using calculators. Scores for 320 CPMP students were compared to those of a control group of 273 students. Computer software was used to create a confidence interval for the difference in mean scores. Confidence level: 95% Variable: Mu(CPMP)-Mu(Control) Interval: (5.573, 11.427) If we had created a 98% CI, would the margin of error be larger or smaller?

Larger

A medical researcher measured the pulse rates of a sample of randomly selected adults and found the following students t-based confidence interval: 95% confidence, 70.887604 < mu(pulse) < 74.497011 If the researcher had calculated a 99% confidence interval, would the margin of error be larger or smaller? Explain.

Larger. More confidence requires more room for error.

Values for the labor force participation rate of women are published by the U.S. Bureau of Labor Statistics. WE are interested in whether there was a difference between female participation in 1968 and 1972, a time of rapid change for women. We check LFPR values for 19 randomly selected cities for 1968 and 1972. Software output follows: Paired t-test of mu(1-2) Test Ho: mu(1972-1968)=0 vs Ha: mu(1972-1962) does not equal 0. Mean of paired differences: 0.0337 t-Statistic: 2.458 with 18 df p= 0.0244 2-sample t-test of mu1-mu2 Ho: mu1-mu2 =0 vs Ha: mu1-mu2 does not equal 0. Test Ho: mu(1972)-mu(1968)=0 vs Ha: does not equal zero. Difference between means: 0.0337 t-Statistic: 1.496 with 35df p= 0.1434 Which of these tests is appropriate for this data and why?

Matched pairs. Same cities in different periods.

Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the mean weight gain this supplement produces has a margin of error of +/- 11 pounds. Some students wrote the following conclusions. Are they correct? Explain. There average weight gain of cows fed this supplement will be between 45 and 67 pounds 95% of the time.

No, the average weight gain of all cows does not vary. It's what we're trying to estimate.

Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the mean weight gain this supplement produces has a margin of error of +/- 11 pounds. Some students wrote the following conclusions. Are they correct? Explain. We're 95% sure that a cow fed this supplement will gain between 45 and 67 pounds.

No, the confidence interval is not for individual cows

Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the mean weight gain this supplement produces has a margin of error of +/- 11 pounds. Some students wrote the following conclusions. Are they correct? Explain. We're 95% sure that the average weight gain among the cows in this study was between 45 and 67 pounds.

No, we know the average gain in this study was 56 pounds.

The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 34% of students had not been absent from school even once during the previous month. In a 2000 survey, responses from 8302 students showed that this figure had slipped to 33%. Officials would, of course, be concerned if student attendance were declining. Do these figures give evidence of a change in student attendance? Do you think this difference is meaningful?

No. A difference is small, even if statistically significant, is probably not meaningful. We might look at new data in a few years.

A catalog sales company promises to deliver orders placed on the internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% +/- 6%. What does this mean? Is this conclusion correct? Explain. 95% of all random samples of customers will show that 88% of orders arrive on time.

Not correct because different samples will give different results.

A catalog sales company promises to deliver orders placed on the internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% +/- 6%. What does this mean? Is this conclusion correct? Explain. On 95% of the days, between 82% and 94% of the orders will arrive on time.

Not correct because the parameter is talking about the interval, not the days.

A catalog sales company promises to deliver orders placed on the internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% +/- 6%. What does this mean? Is this conclusion correct? Explain. Between 82% and 94% of all orders arrive on time.

Not correct because this implies certainty.

A catalog sales company promises to deliver orders placed on the internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% +/- 6%. What does this mean? Is this conclusion correct? Explain. 95% of all random samples of customers will show that 82% to 94% of orders arrive on time.

Not correct because this is about the population proportion, not the sample proportion of different samples.

A catalog sales company promises to deliver orders placed on the internet within 3 days. Follow-up calls to a few randomly selected customers show that a 95% confidence interval for the proportion of all orders that arrive on time is 88% +/- 6%. What does this mean? Is this conclusion correct? Explain. We are 95% sure that between 82% and 94% of the orders placed by the sampled customers arrive on time.

Not correct because we know that 88% of the orders arrive on time with a margin of error of 6%.

Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a 95% confidence interval for the mean weight gain this supplement produces has a margin of error of +/- 11 pounds. Some students wrote the following conclusions. Are they correct? Explain. 95% of the cows studied gained between 45 and 67 pounds.

Not correct. The confidence interval is for the population mean, not the individual cows studied.

What fraction of cars made in Japan? The computer output below summarizes the results of a random sample of 50 autos. Explain carefully what it tells you. z-interval for proportion With 90.00% confidence 0.29938661< P(Japan) < 0.46984416

On the basis of this sample, we are 90 confident that the proportion of Japanese cars is between 29.9% and 47.0%.

Conditions for a proportion hypothesis test

RC, 10%, S/F

The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 34% of students had not been absent from school even once during the previous month. In a 2000 survey, responses from 8302 students showed that this figure had slipped to 33%. Officials would, of course, be concerned if student attendance were declining. Do these figures give evidence of a change in student attendance? Check Conditions and Assumptions.

RC: Students were randomly sampled and should be independent. 10%: 8302 students is less than 10% of the entire student population of the United States. S/F: 34% and 66% of 8302 are > or equal to 10

After getting trounced by your little brother in a children's game, you suspect the die he gave you to roll may be unfair. To check, you roll it 60 times, recording the number of times each face appears. Do these results cast doubt on the die's fairness? Check the conditions.

Rolls are random, data is counts for categorical variable die face, expected frequencies all bigger than 5.

The National Center for Education Statistics monitors many aspects of elementary and secondary education nationwide. Their 1996 numbers are often used as a baseline to assess changes. In 1996, 34% of students had not been absent from school even once during the previous month. In a 2000 survey, responses from 8302 students showed that this figure had slipped to 33%. Officials would, of course, be concerned if student attendance were declining. Do these figures give evidence of a change in student attendance? State your conclusion.

The P-value provides weak evidence against the null hypothesis.

In 2015, the U.S. Census Bureau reported that 62.2% of American families owned their homes- the lowest rate in 20 years. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers to encourage people to become home owners. They decide to adopt a plan on a 2-year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. What's the Type II error?

The city abandons the tax breaks when they were helping increase home ownership

In 2015, the U.S. Census Bureau reported that 62.2% of American families owned their homes- the lowest rate in 20 years. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers to encourage people to become home owners. They decide to adopt a plan on a 2-year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. What's the Type I error?

The city concludes that home ownership is on the rise, but in fact the tax breaks don't help.

Values for the labor force participation rate of women are published by the U.S. Bureau of Labor Statistics. WE are interested in whether there was a difference between female participation in 1968 and 1972, a time of rapid change for women. We check LFPR values for 19 randomly selected cities for 1968 and 1972. Software output follows: Paired t-test of mu(1-2) Test Ho: mu(1972-1968)=0 vs Ha: mu(1972-1962) does not equal 0. Mean of paired differences: 0.0337 t-Statistic: 2.458 with 18 df p= 0.0244 2-sample t-test of mu1-mu2 Ho: mu1-mu2 =0 vs Ha: mu1-mu2 does not equal 0. Test Ho: mu(1972)-mu(1968)=0 vs Ha: does not equal zero. Difference between means: 0.0337 t-Statistic: 1.496 with 35df p= 0.1434 Using the test you selected, state your conclusion.

There is a significant difference (P= 0.0244) in the labor force participation rate for women in these cities; women's participation seems to have increased from 1968 to 1972.

Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Is this statement true? For a fixed margin of error, larger samples provide greater confidence.

True.

Several factors are involved in the creation of a confidence interval. Among them are the sample size, the level of confidence, and the margin of error. Is this statement true? For a specified confidence level, larger samples provide smaller margins of error.

True.

A bank wants to know if the enrollment on their website is above 30% based on a small sample of customers. They test H0: p=0.3 vs Ha: p>0.3 and reject the null hypothesis. Later they find out that actually 28% of customers enrolled. Is this a Type I, Type II, or neither error?

Type I

In 2015, the U.S. Census Bureau reported that 62.2% of American families owned their homes- the lowest rate in 20 years. Census data reveal that the ownership rate in one small city is much lower. The city council is debating a plan to offer tax breaks to first-time home buyers to encourage people to become home owners. They decide to adopt a plan on a 2-year trial basis and use the data they collect to make a decision about continuing the tax breaks. Since this plan costs the city tax revenues, they will continue to use it only if there is strong evidence that the rate of home ownership is increasing. Who gets hurt in Type I and Type II error?

Type I: city forgoes tax break Type II: withdrawals help from those who could've bought a home

A human resource analyst wants to know if the applicants this year score, on average, higher on their placement exam than the 52.5 points the candidates averaged last year. She samples 50 recent tests and finds the average to be 54.1 points. She fails to reject the null hypothesis that the mean is 52.5 points. At the end of the year, they find that the candidates this year had a mean of 55.3 points. Is this a Type I, Type II, or neither error?

Type II

A pharmaceutical company tests whether a drug lifts the headache relief rate from the 25% achieved by the placebo. They fail to reject the null hypothesis because the P-value is 0.465. Further testing shows that the drug actually relieves headaches in 38% of people. Is this a Type I, Type II, or neither error?

Type II

The Core Plus Mathematics Project (CPMP) is an innovative approach to teaching Mathematics that engages students in group investigations and mathematical modeling. After field tests in 36 high schools over a three-year period, researchers compared the performances of CPMP students with those taught using a traditional curriculum. In one test, students had to solve applied algebra problems using calculators. Scores for 320 CPMP students were compared to those of a control group of 273 students. Computer software was used to create a confidence interval for the difference in mean scores. Confidence level: 95% Variable: Mu(CPMP)-Mu(Control) Interval: (5.573, 11.427) Explain what the calculated interval means in this context.

We are 95% confident that students who learned math using CPMP will score on average between 5.57 and 11.43 points higher on a test solving applied Algebra problems.

Using the summary statistics provided in Exercise 11, researchers calculated a 95% confidence interval for the mean difference between Walmart and Target purchase amounts. The interval was (-$14.15, -$1.85). Explain in context what this interval means.

We are 95% confident that the mean purchase amount at Walmart is between $1.85 and $14.15 less than the mean purchase amount at Target.

The information in Exercise 1 was used to create a 95% two-proportion confidence interval for the difference between Canadians and U.S. citizens who were born in foreign countries. Interpret this interval with a sentence in context. 95% confidence interval for: P Canadians- P Americans is (3.24%, 9.56%).

We are 95% confident that the proportion of foreign-born Canadians is between 3.24% and 9.56% higher than the proportion of foreign-born Americans.

Researchers at the national cancer institute released the results of a study that investigated the effect of weed-killing herbicides on house pets. They examined 827 dogs from homes where an herbicide was used on a regular basis, diagnosing malignant lymphoma in 473 of them. Of the 130 dogs from homes where no herbicides were used, only 19 were found to have lymphoma. State an appropriate conclusion.

We are 95% confident that the proportion of pets with a malignant lymphoma in homes where herbicides are used is between 35.6% and 49.5% higher than the proportion of pets with lymphoma in homes where no pesticides are used.

A poll conducted by the University of Montana classified respondents by whether they were male or female and political party. We wonder if there is evidence of an association between being male or female and party affiliation. State a complete conclusion.

With a test statistic of 0.0084 at alpha=0.05, we fail to reject Ho. These data do not provide evidence of a relationship between political party and sex.

The Core Plus Mathematics Project (CPMP) is an innovative approach to teaching Mathematics that engages students in group investigations and mathematical modeling. After field tests in 36 high schools over a three-year period, researchers compared the performances of CPMP students with those taught using a traditional curriculum. In one test, students had to solve applied algebra problems using calculators. Scores for 320 CPMP students were compared to those of a control group of 273 students. Computer software was used to create a confidence interval for the difference in mean scores. Confidence level: 95% Variable: Mu(CPMP)-Mu(Control) Interval: (5.573, 11.427) Does this result suggest that students who learn math with CPMP will have significantly higher mean scores in algebra? Explain.

Yes because zero is not in the interval

Yvon Hopps ran an experiment to determine optimal power and time for microwave popcorn. blah blah blah does a 95% confidence interval suggest that he met his goal of an average of no more than 10% uncooked kernels?

Yes, the confidence interval is from 3.73 to 9.82, falling below 10% desired goal.

Vitamin D, whether ingested as a dietary supplement or produced naturally when sunlight falls on the skin, is essential for strong, healthy bones. The bone disease rickets was largely eliminated in England during the 1950s, but now there is concern that a generation of children more likely to watch TV or play computer games than spend time outdoors is at increased risk. A recent study of 2700 children randomly selected from all parts of England found 20% of them deficient in Vitamin D. Find a 98% confidence interval.

Z*98= 2.326 (t-table bottom row) (18.2%, 21.8%)

homogeneity test

a test comparing the distribution of counts for two or more groups on the same categorical variable

Sales people working vs Sales: a) what can you say about the direction of association? b) what can you say about the form of the relationship? c) what can you say about the strength of the relationship? d) does the scatterplot show outliers?

a)positive b)linear c) strong d)no

A student tests 100 students to determine whether other students on her campus prefer Coke or Pepsi and finds no evidence that preference for Coke is not 0.5. Later, a marketing company tests all students on campus and finds no difference. Is this a Type I, Type II, or neither error?

neither

assumptions and conditions for correlation

quantitative variable condition, straight enough condition, no outliers condition