Ch.10: Significance Tests for a Difference in Proportions

What is the test statistic for a 2-sample z test for a difference in proportions? What does it measure? Is it on the formula sheet?

-it measures where the sample lies in the sampling distribution, and ISN'T on the formula sheet

What are the conditions for a significance tests for the difference of 2 means?

1. independent: 10% condition for both 2. random: random sample for each of the 2 groups of size n1 and n2 in randomized experiment 3. normal

What are the conditions for conducting a 2 sample z test for a difference in proportions? How are these different than the conditions for a 1-sample z interval for p? What do you have to watch for?

1. normal: n1(p1 hat), n1(1 - p1 hat) >= 10 2. random: quote 2 times [can write both are _____ (e.g. SRSs)] 3. independent: 10n <= N and 10n2 <= N2 Different: Use p1 hat not p0 for normal -watch out for tiny letters saying "Assume all conditions are met).

What must you include in the state step for 2 sample intervals?

1. p1= & p2 = (BE SPECIFIC) OR 2, µ1 = & µ2 = (BE SPECIFIC) AND 3. 2 sample z interval for p1 - p2 OR 4. 2 sample t interval for µ1 - µ2 5. CL = ___%

What are the conditions for calculating the 2-sample t interval for µ₁ - µ₂?

1. random: quote 2 times 2. normal: same as for all means 3. independent: 10n1 <= N1 and 10n2 <= N2 (both sample groups themselves and individual observations in each group must be independent)

What are the conditions for calculating a 2-sample z interval for p1-p2?

1. random: quote 2 times or if plural said once -also counts if 1 random sample with individuals from 2 groups & THEN divide 2. normal: n1(p hat) and n1(1 - p1 hat) n2(p2 hat) and n2(1 - p2 hat) >= 10 3. independent: 10n1 <= N1 10n2 <= N2

How are the independent, random, and normal conditions met in an experiment?

1. random:This was a randomized comparative/ controlled experiment. 2. independent: Due to random assignment, these 2 groups of _____ (population) can be viewed as independent. Also, knowing one _________ (population) (e.g. students) gives us no information about another ______ (population's) _________. 3. normal: CLT met or proportion of each's successes and failures is at least 10.

What are the shape, center, and spread of the sampling distribution of x bar 1 - x bar 2?

1. shape: normal if conditions met -sampling distribution of the x bar differences is normal BECAUSE that means -both population distributions are normal -approximately normal if CLT met (BOTH n1 and n2 >= 30)= 2. center: random → center shifts 3. spread: sx instead of σ (SE)

What are the shape, spread, and center of the sampling distribution of p1 hat - p2 hat?

1. shape: normal if conditions met n(p hat) >= 10 n(1 - p hat) >= 10 n2(p hat 2) >= 10 n2(1- p hat2) >= 10 2. center: p1 - p2 = ____ 3. spread: if independent (10n1 <= N1 and 10n2 <= N2) (otherwise computed standard error is larger than true standard error)

What are clues that you are dealing with a matched pairs design?

2 samples of different size - not matched (converse not true) -see if can mix columns: if can shuffle 1 without changing the data, it's NOT matched pairs. If not, it's matched pairs.

What are conditions for conducting a 2-sample t test for µ1 - µ2?

2 times random sample & independent same (10n1 <= N1 and 10n2 <= N2), but for normal: 1. both populations normal 2. both sample sizes >= 30 3. have to do 2 graphs and no obvious skew or outliers

What do you write for the do step for an interval or test?

Depends on the test/ interval: 1. z test: z =, p=, pc= 2. z interval: (#, #) 3. t test: t=, df=, p= 4. t interval: (#, #)

What is the formula for the 2-sample t interval for µ₁ - µ₂? Is it on the formula sheet?

Formula sheet: statistic +- [critical value * standard deviation of statistic] ← ME

How do you know to do an interval vs. a test?

If they want you to do an interval, the word interval will be in there.

What makes it matched pairs vs. 2 separate samples?

Matched pairs: treatments imposed to 2 similar things/ the same thing 2 ways (e.g. similarly intelligent students, post and pre) -2 separate samples: 2 random groups

Can proportions be paired?

No

What is the standard error (SE) of x bar 1 - x bar 2? Is it on the formula sheet? How do you interpret this value?

No, only has σ (not sx) on formula sheet. -SE is on average, how far our sample mean will differ from the population mean difference

What is the standard error of p1-p2? How is this different than the of p1 hat - p2 hat? Why is this different than the standard error we used for significance tests?

SE: doesn't use population data - 2 sample z tests use p hat c (test pool for p)

How can you determine whether there is convincing evidence that something has happened (i.e. there is a difference) using an interval?

See if 0 is in the range of possible values. -If yes: fail to reject null -If no: reject null

What is the pooled (combined) sampling proportion? Why do we pool the sampling proportions?

The pooled sample proportion is everyone in 1 group. We pool the sample proportions because it is our best guess at no difference (p). -test the pool for p (i.e. only proportions pool)

What is important to note about the 2 sample z test and the z interval?

They don't always give consistent results -why?: test statistic used pooled p, while interval uses p1 hat and p2 hat.

How can you conclude causation?

Was it a randomized experiment or an observational study? -experiment: yes -observational: no

How do you conclude 2 sample intervals?

We are ___% confident that the interval from ____ to ___ captures the actual mean (whatever's being measured) for ____ (populations said separately). This interval suggests that the ____ (whatever is being measured) is between ____ and _____ (units) (context).

Is it OK to use your calculator for the Do step? Are there any drawbacks?

Yes, but it's not worth the time to write out the formula.

What is the normal condition for the 2-sample t procedures? Is it more robust than the 1-sample?

Yes, the two-sample t procedures are more robust against non-Normality than the one-sample t methods. 1. Sample size less than 15: Use two-sample t procedures if the data in both samples/groups appear close to Normal (roughly symmetric, single peak, no outliers). If the data are clearly skewed or if outliers are present, do not use t. 2. Sample size at least 15: Two-sample t procedures can be used except in the presence of outliers or strong skewness. 3. Large samples: The two-sample t procedures can be used even for clearly skewed distributions when both samples/groups are large, roughly n ≥ 30.

How do you use intervals to conclude whether there's a significant difference?

Yes/ No. Since 0 is/ isn't in the interval, (the entire interval is positive/ negative) and we subtracted ___ (µ1) - ___ (µ2). It appears that _____ (conclusion).

Why can't you just do n-1 for degrees of freedom?

You could just do the smaller sample size - 1, but the value is conservative since the confidence interval will have a ME as large or LARGER than needed for CL. Accuracy of this method increases with sample size, but calculators are more accurate.

What is meant by "the sampling distribution of the difference between 2 proportions"?

all possible differences between 2 samples -differences in sample proportions is unbiased estimator of difference in population proportions

What about a 2-sample test for a difference in proportions? Why do we pool for this test?

for proportions, we're assuming no difference and our best guess at no difference is a giant proportion

What distribution does the 2 sample t statistic have? Why do we use a t statistic rather than a z statistic? How do you calculate degrees of freedom?

in words: -measures where sample lies on t distribution assuming null is true -complicated df, use technology

randomization distribution

inference about the difference p1 - p2 in the effectiveness of 2 treatments in a completely randomized experiment -when random, normal, and independent conditions are met, usual inference procedures based on the sampling distribution of p1 hat - p2 hat will be approximately correct **in an experiment, we aren't sampling at random for a larger population **approximate vs. actual distributions yield approximate vs. actual p-values (respectively)

What is the formula for the 2-sample z interval for p1-p2? Is it on the formula sheet?

it's on the formula sheet in words: stat +- [critical value * standard deviation of statistic] ← ME

Should you use 2-sample t procedures with paired data? Why not? How can you know which procedure to use?

no, can't rearrange paired data

When doing 2 sample t procedures, should we pool the data to estimate a common standard deviation? Is there any benefit? Are there any risks?

no, there's no t in the pool! Only p!

What can you do in the random condition for a 2 sample z test for p1-p2 instead of just writing it 2 times?

use empty quotes in places where the verbatim words are excluded

What mistake do students often make when defining parameters in experiments? How can you avoid it?

use parameters p and µ -NOT stats x bar and p hat -Ho is ALWAYS Ho= no difference (using <, >, or =/ in Ho is wrong)

What can we do if we somehow know the population standard deviations σ1 and σ2?

we can use a z statistic and the standard Normal distribution to perform probability calculations.

Can the CLT be applied for clearly skewed data for t-tests?

yes

What is the formula for the 2 sample t statistic? Is it on the formula sheet? What does it measure?

µ₁ - µ₂ = 0