unit 9 notes

standard errors are same as...

hypothesis tests

so what's new?

the formulas will change depending on what we are testing the test selection on your TI84 will change

rounding when given data

one more decimal place than the data

when two normal distributions have much overlap:

high noise, lots of overlap (sorta like the image)

confidence interval estimate of p1 - p2

(p hat1 - p hat2) - E < (p1 - p2) < (p hat1 - p hat2) + E the (p1 - p2) in the middle stays as is, and no numbers are substituted in there

q bar

1 - p bar

q hat2

1 - p hat2

A study will be conducted to investigate whether there is a difference in the mean weights between two populations of raccoons. Random samples of raccoons will be selected from each population, and the mean sample weight will be calculated for each sample. Which of the following is the appropriate test for the study?

A two-sample t-test for a difference between population means

2-sample t interval on calculator

STAT TESTS [0]2-SampTInt

A week before a state election, a random sample of voters from City J and a random sample of voters from City K were taken. Of the 100 voters selected from City J, 65 indicated they were supporting a certain candidate for state senate. Of the 125 voters selected from City K, 75 indicated they were supporting the candidate. Which of the following is the correct test statistic for a two-sample z-test for a difference in population proportions for the two cities (J minus K) in their support for the candidate?

(0.65 - 0.60) / √((0.62*0.38)*((1/100)+(1/125)))

when finding critical values or p-values, use the following for determining the number of degrees of freedom (denoted by df). although these two methods typically result in different numbers of degrees of freedom, the conclusion of a hypothesis test is rarely affected by the choice:

1. in the textbook we use the simple and conservative estimate of: df = smaller of n1 - 1 and n2 - 1 2. statistical software packages typically use the more accurate but more difficult estimate given: df = (A+B)^2 / ((A^2 / n1 - 1) + (B^2 / n2 - 1)), where A = s1^2 / n1 and B = s2^2 / n2

Donald believes that western commuters drive an average of 10 miles more per day than eastern commuters do. He selects random samples from each group. The western mean is 23.5 miles, and the eastern mean is 19.4 miles. A 95 percent confidence interval to estimate the difference in population means, in miles, is (2.5,5.7). Which of the following statements is supported by the interval?

Donald is likely to be incorrect because 10 is not contained in the interval.

construct a 95% confidence interval for the difference in the proportion of girls at KHS who prefer to shower in the morning and the proportion of boys who prefer to shower in the morning. 10/59 preferred morning for girls 26/50 preferred morning for boys

E = 0.168 95% CI: -0.519 ≤ p1 - p2 ≤ -0.182 sufficient evidence to support claim that proportions are not equal

Consider a 90 percent confidence interval constructed to estimate the difference between two population proportions. Which of the following is the best interpretation of what is meant by 90 percent confidence?

For repeated random sampling from the populations with samples of the same size, approximately 90% of the confidence intervals constructed will capture the true difference between the population proportions.

Suggest a better way for Mr. Carpenter to use the S.A.T. scores of the 40 students he selected to determine if students in the population perform better on the verbal portion of the S.A.T. than on the math portion.

Mr. Carpenter could do a matched pairs study in which he uses the same students in each sample to see if the scores, on average, were better on the verbal portion. This would not meet the requirements of having two independent samples, and therefore he would have to find the difference in each test score, and then use a one-sample test for the mean.

Surveys were sent to a random sample of owners of all-wheel-drive (AWD) vehicles and to a random sample of owners of front-wheel-drive (FWD) vehicles. The proportion of owners who were satisfied with their vehicles was recorded for each sample. The sample proportions were used to construct the 95 percent confidence interval for a difference in population proportions (FWD minus AWD) for satisfied owners. The interval is given as (−0.01,0.12). A car company believes that the proportion of satisfied owners of AWD vehicles differs from the proportion of satisfied owners of FWD vehicles. Does the confidence interval provide evidence that this belief is plausible?

No. The interval contains 0.

requirements for inferences about two means - independent samples (2-sample t test for estimating population means,when sigma1 and 2 are unknown)

Requirements 1. σ1 and σ2 are unknown. 2. It is not assumed that σ1 = σ2 3. The two samples are independent. 4. Both samples are simple random samples. 5. Each sample comes from a population that is normally distributed or the sample size is greater than 30. Note: one sample could meet one condition and the other sample could meet the other condition.

Researchers studying starfish collected two independent random samples of 40 starfish. One sample came from an ocean area in the north, and the other sample came from an ocean area in the south. Of the 40 starfish from the north, 6 were found to be over 8 inches in length. Of the 40 starfish from the south, 11 were found to be over 8 inches in length. Which of the following is the test statistic for the appropriate test to investigate whether there is a difference in proportion of starfish over 8 inches in length in the two ocean areas (north minus south)?

(0.15−0.275) / √((0.2125*0.7875)*((1/40)+(1/40)))

q hat1

1 - p hat1

Which standard error should be used for hypothesis tests and confidence intervals:

Are σ1 and σ2 known? YES: Use normal distribution with standard error (this almost never occurs in reality) √((σ1^2 / n1) + (σ2^2 / n2)) NO: Can it be assumed that σ1 = σ2? YES: Use t distribution with pooled standard error (some statisticians recommend against this approach) NO: Approximate method: Use t distribution with standard error (use this method unless instructed otherwise) √((s1^2 / n1) + (s2^2 / n2))

Researchers studying two populations of wolves conducted a two-sample t-test for the difference in means to investigate whether the mean weight of the wolves in one population was different from the mean weight of the wolves in the other population. All conditions for inference were met, and the test produced a test statistic of t=2.771 and a p-value of 0.01. Which of the following is a correct interpretation of the p-value?

Assuming that the mean weights of wolves in the populations are equal, the probability of obtaining a test statistic that is greater than 2.771 or less than −2.771 is 0.01.

A study was conducted to investigate whether the mean numbers of snack bars sold at two airport convenience stores, C and D, were different. For ten randomly selected days, the number of snack bars sold at each store was recorded, and the sample mean number of snack bars for each store was calculated. A two-sample t-test for a difference in means will be conducted. Have all conditions for inference been met?

No, the sample sizes are not large enough to assume normality of the sampling distribution.

2-sample z interval on calculator

STAT TESTS [9]2-SampZInt

how to find a 2-sample t test on calculator

STAT [4]2-SampleTTest fill in data pooled: generally no

The critical value is determined using the...

standard normal distribution and a given level of alpha.

A two-sample t-test for a difference in means will be conducted to investigate whether the average amount of money spent per customer at Department Store M is different from that at Department Store V. From a random sample of 35 customers at Store M, the average amount spent was $300 with standard deviation $40. From a random sample of 40 customers at Store V, the average amount spent was $290 with standard deviation $35. Assuming a null hypothesis of no difference in population means, which of the following is the test statistic for the appropriate test to investigate whether there is a difference in population means (Department Store M minus Department Store V) ?

t = (300-290) / (√((40^2 / 35) + (35^2 / 40)))

The standard error for the difference between two means is analogous....

to the standard error of the mean for a single mean.

two samples are dependent if the sample...

values are paired (that is, each pair of sample values consists of two measurements from the same subject (such as before/after data), or each pair of sample values consists of matched pairs (such as husband/wife data), where the matching is based on some inherent relationship.

Researchers conducted an experiment in which people with a certain condition were given either a drug or a placebo to treat the condition. At the significance level of α=0.01, a test of the following hypotheses was conducted. H0: p1 = p2 H1: p1 > p2 In the hypotheses, p1 represents the proportion of all people who experience an allergic reaction while taking the drug, and p2 represents the proportion of all people who experience an allergic reaction while taking the placebo. All conditions for inference were met, and the resulting p-value was 0.12. Which of the following is the correct decision for the test?

The pp-value is greater than αα, and the null hypothesis is not rejected. There is not convincing evidence to support the claim that the proportion of people with an allergic reaction will be greater for those taking the drug than for those taking the placebo.

A 90 percent confidence interval for the proportion difference p1 − p2 was calculated to be (0.247,0.325). Which of the following conclusions is supported by the interval?

There is evidence to conclude that p1 > p2 because all values in the interval are positive.

it is a subtle but important difference between the standard error used in hypothesis testing for two sample proportions and the standard error used in building...

a confidence interval. the former uses p bar and q bar, while the latter uses p hat and q hat. this may lead to slightly different result in our analyses

like most of our test statistics, we compare the...

difference in proportions relative to the standard error of the corresponding measure. if the requirements are met, we can use a z score.

the pooled sample proportion has to be...

in between p hat1 and p hat2

lowering the percent of confidence

makes it more precise and less accurate

like other procedures for statistical inference, hypothesis tests and confidence intervals for two portions must...

meet requirements

two samples are independent if the sample....

values from one population are not related to or somehow naturally paired or matched with the sample values from the other population

A recent newspaper article claimed that more people read Magazine A than read Magazine B. To test the claim, a study was conducted by a publishing representative in which newsstand operators were selected at random and asked how many of each magazine were sold that day. The representative will conduct a hypothesis test to test whether the mean number of magazines of type A the operators sell, μ1, is greater than the mean number of magazines of type B the operators sell, μ2. What are the correct null and alternative hypotheses for the test?

H0: μ1 - μ2 = 0 H1: μ1 - μ2 > 0

A civil engineer tested concrete samples to investigate the difference in strength, in newtons per square millimeter (N/mm2), between concrete hardened for 21 days and concrete hardened for 28 days. The engineer measured the strength from each sample, calculated the difference in the mean strength between the samples, and then constructed the 95 percent confidence interval, (2.9,3.1), for the difference in mean strengths. Assuming all conditions for inference were met, which of the following is a correct interpretation of the 95 percent confidence level?

In repeated samples of the same size, approximately 95 percent of the intervals constructed from the samples will capture the population difference in means.

p bar

aka pooled sample proportion (x1 + x2) / (n1 + n2)

if the requirements are met for a 2-sample z interval,...

this is a z-score corresponding to the level of confidence

test statistic for a two-sample test for proportions

z = ((p hat1 - p hat2) - (p1 - p2)) / (√(((p bar*q bar) / n1) + ((p bar*q bar) / n2))) the null hypothesis is usually p1 = p2 so p1 - p2 generally = 0

always remember to...

list the requirements

x1

number of successes in Sample 1

Degrees of Freedom is complicated for these tests. In our text and others, we use a simple approximation by...

taking the smaller of n1 − 1 and n2 − 1

whenever there is a 0, 1, or 2 that is directly next to a letter/variable, we know that...

the number is in subscript

μσ﹤≤﹥≥≠√⍺

yah

requirements for statistical inferences for proportions for confidence intervals

1. The sample proportions are from two simple random samples that are independent. 2. There must be at least 5 successes and 5 failures in each sample. In symbols, np ≥ 5 nq ≥ 5

A behavioral scientist investigated whether there is a significant difference in the percentages of men and women who purchase silver-colored cars. The scientist selected a random sample of 50 men and a random sample of 52 women who had recently purchased a new car. Of the men selected, 16 had purchased a silver-colored car. Of the women selected, 9 had purchased a silver-colored car. Which of the following is the most appropriate method for analyzing the results?

A two-sample z-test for the difference in population proportions

Is the claim that things are equal or unequal? If yes, do we reject or fail to reject H0? If no, do we reject or fail to reject H0?

Equal: claim goes in null hypothesis: -Equal&reject H0: there is sufficient evidence to reject the claim that... -Equal&fail to reject H0: there is insufficient evidence to reject that claim that... Not equal, less than, or greater than: claim goes in the alternative hypothesis (preferred research design) -No&Reject H0: there is sufficient evidence to support the claim that... -No&fail to reject H0: there is insufficient evidence to support the claim that...

requirements for inferences about two means - independent samples (2-sample z test for estimating population means)

Requirements 1. σ 1 and σ 2 are known. 2. The two samples are independent. 3. Both samples are simple random samples. 4. Each sample comes from a population that is normally distributed or the sample size is greater than 30. Note: one sample could meet one condition and the other sample could meet the other condition.

how to do a hypothesis test for a two-sample test for proportions on the calculator

STAT [6] 2-PropZTests fill in data

how to construct confidence intervals for the difference between two population proportions on a calculator

STAT [9]2-PropZInt fill in data/information

Since the standard deviations of the population are unknown, and it cannot be...

assumed that the standard deviations of the populations are equal, the test statistic is a t with un-pooled variance, provided the other requirements are met.

critical values are the same for...

confidence intervals as they are for hypothesis tests. if it is a = and ≠ test, you have to cut alpha in half and then find the complement to find both (but use the positive CV)

in unit 6, we constructed confidence intervals for population proportions. since we are now comparing sample proportions, it is time to construct...

confidence intervals for the difference between population proportions

test statistic for inferences about two means - independent samples (2-sample z test for estimating population means)

in most cases, the null hypothesis is H0: μ1 = μ2, so μ1 - μ2 = 0

p hat1

x1 / n1

An experiment was conducted to investigate whether there is a difference in mean bag strengths for two different brands of paper sandwich bags. A random sample of 50 bags from each of Brand X and Brand Y was selected. Each bag was held from its rim, and one-ounce weights were dropped into the bag one at a time from the same height until the bag ripped. The number of ounces the bag held before ripping was recorded, and the mean number of ounces for each brand was calculated. Which of the following is the appropriate test for the study?

A two-sample t-test for a difference between population means

Two ride-sharing companies, A and B, provide service for a certain city. A random sample of 52 trips made by Company A and a random sample of 52 trips made by Company B were selected, and the number of miles traveled for each trip was recorded. The difference between the sample means for the two companies (A−B) was used to construct the 95 percent confidence interval (1.86,2.15). Which of the following is a correct interpretation of the interval?

We are 95 percent confident that the difference in population means for miles traveled by the two companies is between 1.86 miles and 2.15 miles.

In a large study designed to compare the risk of cardiovascular disease (CVD) between smokers and nonsmokers, random samples from each group were selected. The sample proportion of people with CVD was calculated for each group, and a 95 percent confidence interval for the difference (smoker minus nonsmoker) was given as (−0.01,0.04). Which of the following is the best interpretation of the interval?

We are 95% confident that the difference in proportions for smokers and nonsmokers with CVDCVD in the population is between −0.01−0.01 and 0.04.

when testing a claim about two population proportions, the P-value method and the traditional method are...

equivalent, but they are not equivalent to the confidence interval method. If you want to test a claim about two population proportions, use the P-value method or traditional method; if you want to estimate the difference between two population proportions, use a confidence interval

μ1 μ2

population mean of population 1 population mean of population 2

σ1 σ2

population standard deviation of population 1 population standard deviation of population 2

s1 s2

sample standard deviation of sample 1 sample standard deviation of sample 1

test statistic for statistical inferences for two proportions (2-sample test for proportions)

z = ((p hat1 - p hat2) - (p1 - p2)) / (√(((p bar*q bar) / n1) + ((p bar*q bar) / n2))) note: the (p1 - p2) is usually 0 because the null hypothesis for these tests is usually H0: p1 = p2

since the standard deviations of the population are known, the test statistic is a...

z, provided the other requirements are met.

point estimate: difference in the sample proportions

(p hat1 − p hat2) = (x1 / n1) − (x2 / n2)

standard error for the difference between population proportions for CONFIDENCE INTERVALS ONLY

√(((p hat1*q hat1) / n1) + ((p hat1*q hat1) / n2))

requirements for statistical inferences for two proportions (2-sample test for proportions) for hypothesis tests

1. the sample proportions are from two simple random samples that are independent. this means that you can't have the same person, place, thing, etc. in both samples. this also means that you can't have naturally paired sample items (like in matched pairs tests). 2. there must be at least 5 successes and 5 failures in each sample. in symbols: np ≥ 5, nq ≥ 5 (or 10). the number 5 is not consistent among all texts. the important point is that if the number of successes or failures is too small, the distribution will be skewed and not well approximated by a normal distribution

standard error for two population means with σ1 and σ2 known

√((σ1^2 / n1) + (σ2^2 / n2))

margin of error for the difference between population proportions

E = z of alpha/2 * (√(((p hat1*q hat1) / n1) + ((p hat1*q hat1) / n2)))

point estimate for the difference between population proportions

p hat1 - p hat2

To test the quality of the math programs at two large high schools, the SAT Math scores for simple random samples of 32 students are recorded. Test the claim that the mean Math SAT score for Kennedy High School is greater than the mean Math SAT score for Lincoln High School. Use α = 0.05

H0 : μ1 = μ2 School 1 = Kennedy School 2 = Lincoln H1 : μ1 > μ2 n1 = n2 = 32 σ1 = σ2 = 100 x1 = 538 x2 = 503 Critical Value: z = 1.645 test statistic: z = = 1.40 p-value 0.0808 fail to reject H0 There is insufficient evidence to support the claim that the mean Math SAT score for Kennedy High School is greater than the mean math score at Lincoln High School.

At many college bookstores, students can decide whether to purchase or to rent a textbook for a class. A study was conducted to investigate whether the percent of rented textbooks for all science classes in the state was greater than the percent of rented textbooks for all literature classes in the state. The following hypothesis test was done at the significance level of α=0.05. H0: p1 = p2 H1: p1 >p2 In the hypotheses, p1 represents the proportion of all science textbooks that are rented, and p2 represents the proportion of all literature textbooks that are rented. All conditions for inference were met, and the resulting p-value was 0.035. Which of the following is the correct decision for the test?

The p-value is less than αα. Since 0.035<0.05, the null hypothesis is rejected, and the claim is supported. There is convincing statistical evidence that the proportion of all science textbooks that are rented is greater than the proportion of all literature textbooks that are rented.

A large company offered gym memberships to its employees as part of a program to keep employees healthy. A random sample of employees with a gym membership and a random sample of employees without a gym membership were taken, and the proportion of employees who had taken at least one sick day in the past month was recorded for each sample. A 90 percent confidence interval for the difference in population proportions (membership minus no membership) was found to be (−0.13,0.05). Employees believe that there is no difference in absenteeism between those with a gym membership and those without a gym membership. Does the confidence interval provide evidence that this belief is plausible?

Yes. The value of 0 is contained in the interval, which indicates that no difference is plausible.

inferences about two proportions include...

confidence intervals and hypothesis tests. these inferences are an extension of our learning from Units 7&8.

test statistic for inferences about two means - independent samples (2-sample t test for estimating population means,when sigma1 and 2 are unknown)

in the numerator there is actually: (x bar1 - x bar2) - (mu1 - mu2) but mu1 - mu2 is generally 0

when two normal distribution have not a lot of overlap:

low noise, not much overlap sorta like the image

for 2-propztests...

make sure you write/specify which is p1 and which is p2

three values needed for a confidence interval for means and proportions

point estimate critical value standard error

Recall from a previous lesson that to test the quality of the math programs at two large high schools, the SAT Math scores for simple random samples of 32 students are recorded. Construct a 90% confidence interval for the difference in mean S.A.T. scores for Kennedy and Lincoln High Schools. School 1 = Kennedy School 2 = Lincoln n1 = n2 = 32 σ1 = σ2 = 100 x bar1 = 538 x bar2 = 503

point estimate: 35 SE: 25.0 CV: 1.645 E = 41.1 90% CI: (-6.1, 76.1)

p1

population proportion for Population 1

p2

population proportion for Population 2

The p-value is equal to the...

probability that a difference greater than or equal to the one observed would be obtained if the difference in the two population means is actually 0. when doing it on a calculator: notice that the results show 55.4 degrees of freedom and a p-value of 0.0606. The inaccuracy of the results obtained using 32 degrees of freedom is minor when using adequate samples.

n1

size of sample from Population 1

Two community service groups, J and K, each have less than 100 members. Members of both groups volunteer each month to participate in a community-wide recycling day. A study was conducted to investigate whether the mean number of days per year of participation was different for the two groups. A random sample of 45 members of group J and a random sample of 32 members of group K were selected. The number of recycling days each selected member participated in for the past 12 months was recorded, and the means for both groups were calculated. A two-sample t-test for a difference in means will be conducted. Which of the following conditions for inference have been met? 1. The data were collected using a random method. 2. Each sample size is less than 10 percent of the population size. 3. Each sample size is large enough to assume normality of the sampling distribution of the difference in sample means.

1 and 3 only

Mr. Carpenter was investigating the performance of Kennett students on the verbal and math portions of the S.A.T. He randomly selected 20 students and recorded their verbal scores and 20 other students and recorded their math score. Mr. Carpenter determined that the means of the samples were 527 for verbal and 516 for math. Verbal and math scores are normally distributed. The population standard deviation for each part of the S.A.T. is 100. 1. Test the claim that, on average, students at Kennett perform better on the verbal portion of the S.A.T. than they do on the math portion. Include requirements, hypotheses, a p-value, and a statistical conclusion. 2. What is the point estimate in this situation? 3. What is the standard error in this situation? 4. Interpret the p-value.

1. All of the requirements were met, as both 𝞂1 and 𝞂2 are known as 100, the two samples are independent from one another and are random samples, and the distribution of the samples are normally distributed. 𝝻1 = the population mean of the score on the verbal portion of the SAT 𝝻2 = the population mean of the score on the math portion of the SAT H0: 𝝻1 = 𝝻2 H1: 𝝻1 ﹥ 𝝻2 z = 0.35, Critical Value: 1.645, Alpha = 0.05, P-value = 0.364 Since the P-value of 0.364 is greater than the alpha level of 0.05, we fail to reject H0. There is insufficient evidence to support the claim that, on average, students at Kennett perform better on the bernal portion of the SAT than they do on the math portion. 2. 11 3. 31.62 4. There is a probability of 0.364 that a difference of two sample means of at least 11 would be obtained if the population means are equal.

A yearbook company was investigating whether there is a significant difference between two states in the percents of high school students who order yearbooks. From a random sample of 150 students selected from one state, 70 had ordered a yearbook. From a random sample of 100 students selected from the other state, 65 had ordered a yearbook. Which of the following is the most appropriate method for analyzing the results?

A two-sample z-test for a difference in population proportions

A potato chip company produces a large number of potato chip bags each day and wants to investigate whether a new packaging machine will lower the proportion of bags that are damaged. The company selected a random sample of 150 bags from the old machine and found that 15 percent of the bags were damaged, then selected a random sample of 200 bags from the new machine and found that 8 percent were damaged. Let pˆO represent the sample proportion of bags packaged on the old machine that are damaged, pˆN represent the sample proportion of bags packaged on the new machine that are damaged, pˆC represent the combined proportion of damaged bags from both machines, and nO and nN represent the respective sample sizes for the old machine and new machine. Have the conditions for statistical inference for testing a difference in population proportions been met?

All conditions for making statistical inference have been met.

A random sample of monarch butterflies and a random sample of swallowtail butterflies were selected, and the difference in the average flying speed for each sample was calculated. A two-sample t-test for the difference in means was conducted to investigate whether the speed at which monarchs fly, on average, is faster than the speed at which swallowtails fly. All conditions for inference were met, and the p-value was given as 0.072. Which of the following is a correct interpretation of the p-value?

Assuming that monarchs and swallowtails fly at the same speed on average, the probability of observing a difference equal to or greater than the sample difference is 0.072.

Students at Hereford High School want to investigate whether they have more school spirit than students at Blake High School. To test this hypothesis, the students will select a random sample of students from each school and determine the proportion of the sampled students who wear school colors to their respective pep rallies. Let p1 represent the proportion of Hereford students who wear school colors to their pep rally and p2 represent the proportion of Blake students who wear school colors to their pep rally. Which of the following are the correct null and alternative hypotheses for the investigation?

H0: p1 - p2 = 0 H1: p1 - p2 > 0

A farmer wants to investigate whether a new pesticide will decrease the proportion of pumpkin plants that are being eaten by bugs in the farmer's pumpkin patches compared to the current pesticide being used. The farmer applied the old pesticide to patch A and the new pesticide to patch B. Let p1 represent the proportion of pumpkin plants eaten by bugs in patch A and p2 represent the proportion of pumpkin plants eaten by bugs in patch B. Assume all conditions for inference were met. Which of the following are the correct null and alternative hypotheses to test whether the new pesticide results in fewer pumpkin plants eaten by bugs?

H0: p1 = p2 H1: p1 > p2

A group of AP Chemistry students debated which fast-food chain had better quality bags, Fast Food Chain W or Fast Food Chain M . They decided to investigate by selecting a random sample of 25 bags from each fast food restaurant, slowly adding water until each bag began to leak, and recording the volume of water they were able to pour into each bag. They then calculated the mean volume and standard deviation, in ounces, for the two types of bags. Which of the following are the correct null and alternative hypotheses to test whether the mean volume of water the bags from Fast Food Chain W can hold without leaking, μ1, is different from that for the bags from Fast Food Chain M, μ2 ?

H0: μ1 - μ2 = 0 H1: μ1- μ2 ≠ 0

Hannah claims that people who live in southern states spend 9 hours more per week outside than do people in northern states. She selects a random sample from each group. The mean number of hours per week that people in southern states spent outside is 18.6, and the mean number of hours per week that people in northern states spent outside is 14.4. A 99 percent confidence interval to estimate the difference in population means (southern minus northern) is (0.4,8.0). Which of the following statements about Hannah's claim is supported by the interval?

Hannah is likely to be incorrect because 9 is not contained in the interval.

Medical researchers are studying a certain genetic trait found in two populations of people, W and X. From an independent random sample of people taken from each population, the difference between the sample proportions of people who carried the trait (W minus X) was 0.22. Under the assumption that all conditions for inference were met, a hypothesis test was conducted using the following hypotheses. H0: p1 = p2 H1: p1 > p2 The p-value of the test was 0.03. Which of the following is the correct interpretation of the p-value?

If the proportions of all people who carry the trait are the same for both populations, the probability of observing a sample difference of at least 0.22 is 0.03.

Independent random samples of students were taken from two high schools, R and S, and the proportion of students who drive to school in each sample was recorded. The difference between the two sample proportions (R minus S) was 0.07. Under the assumption that all conditions for inference were met, a hypothesis test was conducted where the alternative hypothesis was the population proportion of students who drive to school for R was greater than that for S. The p-value of the test was 0.114. Which of the following is the correct interpretation of the p-value?

If the proportions of all students who drive to school are the same for both high schools, the probability of observing a sample difference of at least 0.07 is 0.114.

A research group studying cell phone habits asked the question "Do you ever use your cell phone to make a payment at a convenience store?" to people selected from two random samples of cell phone users. One sample consisted of older adults, ages 35 years and older, and the other sample consisted of younger adults, ages 18 years to 34 years. The proportion of people who answered yes in each sample was used to create a 95 percent confidence interval of (0.097,0.125) to estimate the difference (younger minus older) between the population proportions of people who would answer yes to the question. Which of the following is the best description of what is meant by 95 percent confidence?

In repeated random sampling with the same sample size, approximately 95% of the intervals constructed from the samples will capture the difference in population proportions of people who would answer yes to the question.

Two schools are investigating whether there is a difference in the proportion of students who attend the homecoming football game. Both schools have over 2,000 students. School A selected a simple random sample of 100 students and found that 98 attended the homecoming football game. School B selected a simple random sample of 150 students and found that 142 attended the homecoming football game. Let phat3 represent the combined sample proportion for the two schools, and let n1 and n2 represent the respective sample sizes. Have the conditions for statistical inference for testing a difference in population proportions been met?

No, the condition that the distribution of the difference in sample proportions is approximately normal has not been met, because n1(1−p hat3) is not greater than or equal to 5.

in an informal study conducted by AP stats students it was found that 10 out of 59 female students preferred showering in the morning while 26 out of 50 students preferred showering in the morning. test the claim that the percent of female students at KHS who prefer showering in the morning is not equal to the percent of male students at KHS who prefer showering in the morning Use ⍺ = 0.05.

Note: we know the samples were not simple random samples and the samples were more than 5% of the population. However, we will proceed with the analysis as an academic exercise. The second requirement: np1 = 10 ≥ 10, nq1 = 49 ≥ 10, np2 = 26 ≥ 10, nq2 = 24 ≥ 10 H0: p1 = p2 H1: p1 ≠ p2 n1 = 59 n2 = 50 x1 = 10 x2 = 26 p hat1 = 0.169 p hat2 = 0.52 p bar = 0.330 q bar = 0.670 z = -3.88 (unusual, beyond chance variation) critical value: invNorm(0.95,0,1,CENTER) = +/-1.96 reject H0 sufficient evidence to support claim that...

A two-sample t-test for a difference in means was conducted to investigate whether there is a statistically significant difference in the average amount of fat found in low-fat yogurt and the average amount of fat found in nonfat yogurt. With all conditions for inference met, the test produced a test statistic of t=2.201 and a p-value of 0.027. Based on the p-value and a significance level of α=0.05, which of the following is the correct conclusion?

Reject the null hypothesis because p < α. The difference in the average amount of fat found in low-fat and nonfat yogurt is statistically significant.

A survey of randomly selected students and parents included the question: Should schools be required to reduce highly processed foods in the cafeteria? The results are shown below: Students/yes: 135 parents/yes: 78 total students: 220 total parents: 109 Construct a 95% Ci for the difference in proportions (students - parents)

Requirements 1. The students and parents were randomly selected. 2. There are at least 5 students and 5 parents who answered yes, and at least 5 students and 5 parents who did not answer yes. A two-proportion z-interval can be used. Point estimate: -0.1020 CV: z = 1.96 SE: 0.05427 E: 0.1064 −0.208 < p1 − p2 < 0.004 Note that 0 is included in the confidence interval. The results suggest that it is reasonable to assume that differences between proportions of students and parents who answered yes to the survey question can be attributed to chance variation. Note that a more precise CI (like 90%) would suggest otherwise.

An experiment is conducted to study the effect of peer tutoring on reading. Two groups were randomly selected and each provided with 6 weeks of instruction. The experimental group was provided with peer tutoring in addition to instruction while the students in the control group had no peer tutoring. A summary of the results on the reading tests administered at the end of six weeks is displayed in the table below. Experimental Group: n = 32, x bar = 368.4, s = 39.5 Control Group: n = 32, x bar = 349.2, s = 56.6 Let's designate the Experimental Group as Group 1 and the Control Group as Group 2. Test the claim that (in a population with similar characteristics to students in the sample) students who receive peer tutoring will have higher reading scores on average. Use alpha = 0.05

Requirements 1. We don't know the standard deviations of the populations. 2. We don't assume that the standard deviations of the populations are normal. 3. There are two different groups of readers, so the samples are independent. 4. The groups were randomly selected. 5. There are more than 30 students in each group. H0: μ1 = μ2 H1: μ1 > μ2 The degrees of freedom = 31 since both groups have 32 students. The critical value is invT(0.95,31) = 1.696 The point estimate is 19.2 SE: 12.20 test statistic: t = 1.57 P-value can be obtained using tcdf (lower:1.574, upper:100, df:31) = 0.0628 Since the test statistic is not in the critical region, fail to reject the null hypothesis. There is insufficient evidence to support the claim that students who receive peer tutoring have higher reading scores on average.

To test the effectiveness of sending reminder notices to voters, a local town official randomly selected 100 people to send reminders about voting day through the postal service and 100 people as a control group. 68 people who received reminders voted on Election Day. Only 43 people in the control group voted on Election Day. Test the claim that the proportion of people who voted was greater among the people who received reminders than the corresponding proportion in the control group. Use α = 0.05

Requirements for test of difference between two proportions are met: 1. Both samples are simple random samples that are independent of one another. 2. There are at least 5 voters and nonvoters in each sample. p1 = the proportion of people who received reminders that voted. p2 = the proportion of people in the control group that voted. H0 : p1 = p2 H1 : p1 > p2 The claim is in the alternative hypothesis. Critical Value: z = 1.645 test statistic: z = 3.57 The p-value can be found using a normal cdf with lower bound = 3.57, upper bound = 100, mean = 0, and standard deviation = 1. P-value = 1.79*10−4 Since the p-value is less than alpha, reject H0 Since z = 3.57 is in the critical region, we reject H0. There is sufficient evidence to support the claim that the proportion of people who voted was greater among the people who received reminders than the corresponding proportion in the control group.

how to do a hypothesis test for two means when population standard deviation is known

STAT [3] 2-SampleZTest enter in data/info

Uncle Jack is trying to determine if equal proportions of students in his Block 2 and Block 4 classes accepted invites to Google Classroom. In Block 2, 14 out of 22 students accepted the invitation. In Block 4, 7 out of 11 students accepted the invitation. Explain why the requirements for a z test for the difference in two proportions are not met.

The requirements for a z test for the difference in two proportions are not met because these are not two simple random samples, but are two censuses of the two classes. These two samples are independent of each other, as there are different students in the two classes. Also, although there are at least 5 successes and 5 failures in block 2, there are not 5 failures in block 4, so this requirement is not met.

In Cornway, a mean pollution index of 43 with a standard deviation of 10 was found for a random sample of 12 days. In nearby Hamburg, a mean pollution index of 36 with a standard deviation of 15 was found for a random sample of 14 days. Test the claim that the mean pollution index in Hamburg is less than the mean pollution index in Cornway. Assume that the pollution index in each town is approximately normal.

The requirements have been met as the population standard deviations are unknown, it is not assumed that 𝞂1 = 𝞂2 since 𝞂1 and 𝞂2 are unknown, both samples are independent of one another, both samples were selected randomly, and the pollution index in each town is said to be approximately normal. 𝝻1 = the population mean of the pollution index found in Cornway 𝝻2 = the population mean of the pollution index found in Hamburg H0: 𝝻1 = 𝝻2 H1: 𝝻1 ﹥ 𝝻2 Point Estimate: 43 - 36 = 7 Standard error: √(102/12) + (152/14) = 4.94 Test Statistic = 7 / 4.94 = 1.42 Degrees of Freedom: 11 Critical Value: invT(0.95,11) = 1.80 Since the test statistic is not in the critical region, we fail to reject H0. There is insufficient evidence to support the claim that the mean pollution index in Hamburg is less than the mean pollution index in Cornway.

Sam is contemplating moving to Boulder, Colorado after his senior year so that he can enjoy outdoor sports like mountain biking and Nordic skiing. After studying Unit 9 in his AP Stats course, Sam decides to take a random sample of 75 winter days in Boulder and 60 winter days in Conway. The samples were taken from the last 10 years. Sam found that it snowed 17 out of 75 winter days in Boulder and 22 out of 60 winter days in Conway. Construct a 97% confidence interval for the difference in proportions of snow days in Boulder and Conway. Show all relevant information: requirements, critical value, standard error, and E.

The requirements have been met as these are both random samples that are independent of one another, and there are at least 5 successes and 5 failures in each sample. p hat1 = the proportions of days it snowed in Boulder p hat2 = the proportion of days it snowed in Conway Point estimate = p hat1 - p hat2 = -0.14 Critical Value: +/-2.17→CV = 2.17 Standard Error = 0.0788 E = 2.17 * 0.0788 = 0.171 97% Confidence Interval: -0.311 ﹤ p1 - p2 ﹤ 0.0310 Since 0 is included in the confidence interval, the results suggest that it is reasonable to assume that the differences between the proportions of the days it snowed in Boulder and Conway can be attributed to chance variation.

In a simple random sample of 50 students at a large university in the fall semester, 32 earned a grade of C or higher. In a simple random sample of 52 students in the spring semester, 40 earned a grade of C or higher. Test the claim that a smaller proportion of students earned a grade of C or higher in the fall. Use alpha = 0.10. Your analysis should include: Requirements, Null and Alternative Hypotheses, a test statistic and a critical value, and a statistical conclusion.

The requirements have been met, as the samples are independent of each other and are simple random samples, and there are at least 5 successes and 5 failures in each sample. p1 = the proportion of students who earned a grade of C or higher in the fall semester p2 = the proportion of students who earned a grade of C or higher in the spring semester H0: p1 = p2 H1: p1 < p2 CV: z= -1.28 Test Statistic: z = -1.43 Since z = -1.43 is in the critical region, reject H0. There is sufficient evidence to support the claim that a smaller proportion of students earned a grade of C or higher in the fall.

Two 95 percent confidence intervals will be constructed to estimate the difference in means of two populations, R and J. One confidence interval, I(subscript)400, will be constructed using samples of size 400 from each of R and J, and the other confidence interval, I(subscript)100, will be constructed using samples of size 100 from each of R and J. When all other things remain the same, which of the following describes the relationship between the two confidence intervals?

The width of I(subscript)400 will be 0.5 times the width of I(subscript)100.

A consumer group selected 100 different airplanes at random from each of two large airlines. The mean seat width for the 100 airplanes was calculated for each airline, and the difference in the sample mean widths was calculated. The group used the sample results to construct a 95 percent confidence interval for the difference in population mean widths of seats between the two airlines. Suppose the consumer group used a sample size of 50 instead of 100 for each airline. When all other things remain the same, what effect would the decrease in sample size have on the interval?

The width of the confidence interval would increase.

A two-sample t-test for a difference in means was conducted to investigate whether the average wait time at a fast food restaurant in Town A was longer than the average wait time at a fast food restaurant in Town B. With all conditions for inference met, the test produced a test statistic of t=2.42 and a p-value of 0.011. Based on the p-value and a significance level of α=0.02, which of the following is a correct conclusion?

There is convincing statistical evidence that the average wait time at the restaurant in Town A is longer than the average wait time at the restaurant in Town B.

A basketball coach is experimenting with different techniques for teaching foul shooting. She videotapes Jaelin shooting 50 foul shots, of which Jaelin makes 38. The coach then provides instruction on proper follow through. After the instruction, Jaelin makes 45 out of 50 shots. The coach then uses a z-interval for the difference in two proportions to estimate how much of a difference her instruction made. Give two reasons why this situation does not meet the requirements for a z interval.

This is not two simple random samples because it is only one person in the sample. The samples are also not independent of one another, because it has the same person shooting in each sample, as this is a matched pairs study.

more information =

more precise estimate

as in unit 8, we establish...

null and alternative hypotheses, then test the null hypothesis using either... method 1: a test statistic and a critical value(s) or method 2: a p-value and an alpha level

x2

number of successes in Sample 2

if the p-value is less than alpha,... if the p-value is greater than alpha,... if the test statistic falls within the critical region,... if the test statistics falls outside the critical region,...

reject H0 fail to reject H0 reject H0 fail to reject H0

maximum error is the

same as proportion confidence intervals E= CV*SE

rounding when given summary statistics

same as summary statistics

what is necessary to do:

set up the correct analysis interpret the results appropriately

n2

size of sample selected from Population 2

As you might expect, the t-interval will be...

slightly less precise than the corresponding z-interval.

We construct z intervals in the rare case that the... We construct t intervals in the usual case that the...

standard deviations of both populations are known. population standard deviations are unknown.

now that we have the gist of hypothesis testing, going forward involves applying the same type of thinking to new situations. the practice of finding and interpreting a p-value...

stays the same, as does the traditional method of finding a test statistic and comparing the test statistic to a critical value(s)

A two-sample t-test will be conducted to investigate whether the mean number of tickets sold for children each day is less at movie theater J than at movie theater K. From a random sample of 50 days at theater J, the average was 75 children tickets with standard deviation 12. From a random sample of 60 days at theater K, the average was 85 children tickets with standard deviation 14. Under the assumption that there is no difference in the population means (J minus K), which of the following is the appropriate test statistic for the test?

t = (75-85) / (√((12^2/50)+(14^2/60)))

As with one-sample tests for the mean, when the standard deviations of the population are unknown, we use a...

t-distribution. Note that the standard error looks just like the standard error from our two-sample z test, except that σ1 and σ2 have been replaced by s1 and s2 respectively

requirements for a two-sample test of proportions

the sample proportions are from two simple random samples that are independent. samples are independent if the sample values selected from one population are not related to or naturally paired or matched with the sample values selected from the other population. for each of the two samples, the number of successes is at least 5 and the number of failures is at least 5 (or 10: np ≥ 10, np ≥ 10)

be sure to...

verify all requirements!!!!!!!!!!! and check both test statistic, p-value, and confidence interval

confidence interval for z and t tests for two means

x bar1 − x bar2 − E < μ1 − μ2 < x bar1 − x bar2 + E μ1 − μ2 stays how it is

p hat2

x2 / n2

point estimate for both z and t intervals

xbar1 - xbar2

for terms of this stats: we say that

σ1 does not equal σ2 if we do further and dive in, we may say yes it does equal

standard error for a two-sample test for proportions for HYPOTHESIS TESTS ONLY

√(((p bar*q bar) / n1) + ((p bar*q bar) / n2))