Quiz #2

A national survey of restaurant employees found that 75% said that work stress had a negative impact on their personal lives. The owner of the national chain of Cool Stat Cafe is wondering whether the percentage differs from her restaurants. A random sample of 100 of her employees finds that 68 answer "yes." Her null and research hypotheses are H₀: p = .75; H₁: p ≠ .75. Conduct the test and state your conclusion. For the above situation, give a 95% confidence interval for p. Is your answer compatible with the conclusion you gave in #1?

1 Prop Z test H₀: p = .75 H₁: p ≠ .75 n = 100 x =68 p = .11 fail to reject 1 Prop Z Interval (.58857, .77143) This supports the claim because .75 is within the interval, therefore we would fail to reject. We are 95% sure the data is within this interval and it is.

H₀: p = .1 H₁: p ≠ .1 n= 82 number of successes: 13 Find the P-value Find a confidence interval Is the confidence interval compatible with our answer?

1 prop Z test: P₀ = .1 x = 13 n = 82 p = .08 (fail to reject) 1 prop Z interval: (.07948, .23759) This supports our claim. .1 falls within this interval. Therefore we would fail to reject

When the margin of error is +/- .03, what is our sample size?

1,067

Steps in calculator to test paired data

1. Input data into appropriate lists 2. Subtract L₂ from L₁ and store results in L₃ 3. Go to T/Z Test and use data with L₃ as the list and frequency at 1 (to do the interval) 1. Go to T/Z interval and enter L₃

An airline credit card company runs a pilot promotion where they give double miles for a year to a simple random sample of 25 customers. They are interested to know whether the promotion leads to an increase in the average amount that customers charge on the card. For the 25 customers, they find that the mean increase is $153 and the sample standard deviation is $49. 1. State the null and alternative hypotheses. Do you think a one-tailed test would be justified in this situation? 2. Is there significant evidence at the 1% level that the population mean amount charged increases under the promotion? 3. Recall that the presence of outliers violates the assumption of a t-test. Explain why the credit limit enforced on each card might prevent extreme outliers.

1. µ₁ : card; µ₂ : no card H₀: µ₁ = µ₂ H₁: µ₁ > µ₂ A one-tailed test is justified because if they have more miles, they are more likely to use them. 2. Yes. H₀: µ = 0 ; H₁: µ > 0 α = .01. T test: µ = 0; x bar = 153; Sx = 49; n = 25; µ > 0 p = 2.26^-14. reject 3. The customers cannot go beyond the limit so there cannot be outliers.

Cesar collects data on bicycle accidents, alcohol and gender. He wants to know whether among accident victims, there is a difference in proportions for each gender that tests positive for alcohol. Male; n = 1520; x = 330 Female; n = 191; x = 27 A) Give a 95% confidence interval for the difference of the population proportions Pmale - Pfemale B) Perform a test of hypotheses: H₀: Pmale = Pfemale H₁: Pmale ≠ Pfemale C) are your answers to parts A and B compatible?

A) 2 prop Z interval (.02216, .12932) B) 2 prop Z test p = .015 (reject) C) They are compatible because 0 is not in the interval, therefore we would reject.

Which design? The following situations all require interference about a mean or means. Identify each as 1) single sample 2) matched pairs 3) two independent samples A) Your customers are college students who either live in a dorm or off-campus. You are interested in comparing the interest of these two groups in a new product you are developing. B) Your customers are college students. You are interested in comparing which of two new products you are developing would be more attractive to your customers. C) You want to estimate the average age of your customers. D) You do a survey of your customers every year. One of the questions in the survey asks about customer satisfaction on a seven-point Likert scale (where 1 means very dissatisfied and 7 means very satisfied). Has the mean customer satisfaction improved from last year to this?

A) independent B) matched C) single D) matched

Men: n = 23 x bar = 138 S₁ = 11.5 Women: n = 21 x bar = 143 S₂ = 13.6 Determine whether you can pool the data or not using the appropriate test. Run the appropriate test given α = .05

F Test You can pool this data. P = .44 Fail to reject 2Sample T Test P = .19 Fail to reject

Does cocaine use by pregnant women cause their babies to have low birth weight? Here is the data: Cocaine positive: n = 134; x bar = 2733; s = 599 Cocaine negative: n = 5974; x bar = 3118; s = 672 Perform a one-tailed test with α = .05. Compute Cohen's d and talk about the effect size.

F test: p = .08 (fail to reject, we can pool) µ₁ < µ₂ 2 sample T test: p = 2.66E-11 (reject) Cohen: d = abv(2733 - 3118) / 670.494604 d = .5742 (medium effect)

Before: 40, 38, 95, 111, 120, 162 After : 52, 43, 99, 98, 120, 170 Did this test make a difference in mean strength? State both hypotheses Run the appropriate test with α = .05 Give a 95% confidence interval. Is this interval compatible with our answer?

H₀ = µ H₁ ≠ µ T Test P = .48 (fail to reject) T Interval CI: (-6.426, 11.759) This interval is compatible because 0 falls within the interval. Therefore we fail to reject

How accurate are radon detectors of a type of sold to homeowners? To answer this question, university researchers placed 12 detectors in a chamber that exposed them to 105 picocuries per liter of radon. The detector readings were as follows: 91.9, 97.8, 111.4, 122.3, 95.0, 103.8, 99.6, 96.6, 119.3, 104.8, 101.7. Is there convincing evidence at the .05 level that the mean reading of all detectors of this type differs from the true value 105?

H₀: µ = 105 H₁: µ ≠ 105 T-Test p = .74 (fail to reject) There is not convincing evidence

As you increase your sample size, does your data get wider or narrower?

Narrower

If P < α, we _____ If P > α, we _______

P < α : reject P > α : fail to reject

How do we find P hat? How do we find Q hat

P hat: number of successes / total sample size Q hat: 1 - P hat

When there is a two tailed test, what happens to the P Value? Does this lead to more or less rejections?

P value doubles Leads to less rejections

What is paired data? What is independent data?

Paired data means there is a link between the data. Key word is 'same' Independent data means there is no link. There is usually some form of a 'control' group and a 'experiment' group

What are the two techniques we can use with independent data?

Pooled (only if σ₁ is close to σ₂) Satterwaite (when σ₁ is not close to σ₂)

Suppose Leslie wants to see if the daily number is fixed. She played every night during 2015, betting on the number 314 each time. She won once. Is this a case where she should or should not rely on the calculator to give a confidence interval for the true probability of winning? Defend your answer.

She should not rely on the calculator. The calculator uses an algorithm that is only accurate when the sample has at least 5 successes and 5 failures. She only had 1 success and therefore the information she gets from the calculator will not be reliable.

The table below gives the pretest and posttest scores on the MLA listening test in Spanish for 20 executives who received intensive training in Spanish. Pre: 30, 28, 31, 26, 20, 30, 34, 15, 28, 20, 30, 29, 31, 29, 34, 20, 26, 25, 31, 29 Post: 29, 30, 32, 30, 21, 28, 36, 16, 29, 25, 28, 28, 31, 32, 32, 22, 28, 27, 32, 29 Carry out a test to see if the training had a positive impact. Can you reject at the 5% level? At the 1% level? Would your answers have changed had you conducted a two-tailed test?

T interval: At 95%: (.0456, 1.8544) 0 is not in the interval, therefore we reject At 99%: (-.2862, 2.1862) 0 is in the interval, therefore we will fail to reject Had we preformed a two-tailed test, our p value would have doubled.

What does Cohen's theory test?

Tests effect size (d) It attempts to see if the difference in means is enough to care about.

In paired data, there is usually a 'before and after' that we compute the difference from. What is another name for this difference?

The 'Gain Score'

How do we determine whether we can pool our data or not? How do we use this test?

Use the "F" test. Go into your calculator and use the "F' test. Enter the data. P < α : reject (cannot pool) P > α : fail to reject (can pool)

What to look for with paired data?

Usually has a 'before and after'

When do we use the calculator to test a proportion?

We should only rely on the calculator when there has been 5 successes and 5 failures.

When 0 is in the confidence interval, we __________ When 0 is not in the confidence interval, we _________

Within the interval: fail to reject Not in the interval: reject

Which is narrower? 'Z' intervals or 'T' intervals?

Z intervals

What test do you use with proportions?

Z test

When do we use a Z test? When do we use a T test?

Z test: when σ is known T test: when σ is not known

Cohen Equation

abv(xbar1 - xbar2) / Sp

Cohen's test (what numbers determine effect size)

d ≤ .2 : small effect d ≈ .5 : medium effect d ≥ .8 : large effect

What do the hypotheses look like with tests concerning one mean?

µ =/>/</≠ any number

What do the hypotheses look like with tests concerning two means?

µ₁ =/>/</≠ µ₂

We compare the difference of means to see how...

...spread out the data is

When you pool data, you are assuming...

...the variances are the same