Chapter 10

Compare the standard deviations of the three distributions. Why do you think the SD of Thumb is so much larger than the SD of the two sampling distributions? (Check all that apply.)

In general, means vary less than individuals, because a bunch of means tend to cluster around the center. Large samples (e.g., n=157) will tend to have individuals from the lower, middle, and upper end of the range of measurements. Means of such samples will end up in the same place (the middle) because they generally balance very long and very short thumbs in the sample.

𝑏1

Increment (in mm) added to Thumb for each additional one inch of Height

What varies more, individual scores or sample means?

Individual scores

If we put in .975, so that our code would be xqt(.975, df = 999), do you think we would get the same critical t score?

It would be almost the same—the critical t score would just be positive because it will be on the right side of the 0.

If we needed only 90% confidence, would our confidence interval be narrower or wider?

Narrower If you don't need to be as confident, your confidence interval doesn't need to cover as many possible values. The 90% likelihood area is smaller, so the lower bound sampling distribution would need to move up, and the upper bound sampling distribution would need to move down, to put the sample estimate on the unlikely boundary. The upper and lower bound sampling distributions would both move in closer to keep the sample mean on the unlikely boundary.

If we simulated this sampling distribution of means again, would we get exactly the same numbers for the lower and upper cutoff? Why or why not?

No, because simulations involve randomly generating samples so we won't generate exactly the same samples. But if we simulate a large number of samples (1,000), the lower and upper cutoffs will be close.

Compare the original data (TinyFingers$Thumb) with the new sample of data that results from resample(). Is the resampled data set exactly the same as TinyFingers$Thumb? (Check all that apply.)

No, some of the numbers from the original data appear multiple times in the resampled data. No, some of the numbers from the original data are not in the resampled data.

From our simulated confidence interval, we can also take a look at our margin of error. The distance between the upper bound and the mean of our sample (61.51 - 60.1) is 1.41. Previously, when we first simulated confidence intervals, we got a margin of error around 1.39. Why didn't we get the exact same margin of error in both sets of simulations?

Random simulations have an element of chance. Each simulation will have some fluctuation.

The graph with one peak

Range of thumb lengths that correspond to the middle 95% of individuals in the population

From our bootstrapped confidence interval, we can also take a look at our margin of error. The distance between the upper bound and the mean of our sample (61.4 - 60.1) is 1.3. Why didn't we get the exact same margin of error from bootstrapping and simulations?

Resampling and simulations have an element of chance. Each random sample generated with these processes will have some fluctuation.

From all of our explorations of sampling distributions so far, what do the bootstrapped and simulated sampling distributions have in common? (Check all that apply.)

Shape Center (e.g., mean) Spread (e.g., standard deviation)

The graph with two peaks

The confidence interval of means: the range of population means from which our estimate would be considered "likely"

What would a "slightly bigger" confidence interval mean? (Check all that apply.)

The lower bound (2.5%) value would be smaller. The upper bound (97.5%) value would be bigger.

Which distribution do the numbers in the confidence interval represent? For example, if we are trying to estimate the confidence interval of the mean, which distribution's mean are we 95% confident lies in this range?

The population

If you wanted to know what the range within which 95% of individual scores would fall, which distribution would you need to create a model of?

The population: the entire distribution of individuals

Which are accurate explanations of the 95% confidence interval of thumb length? (Check all that apply.)

The range of parameter values from which our estimate would be considered "likely" to have come from The range of population means from which our estimate would be considered "likely".

What is the correct interpretation of the margin of error?

The range of possible DGPs above and below our parameter estimate that would be likely to have produced our estimate

Why do you think the mean of the simulated sampling distribution is the same as the mean of the bootstrapped sampling distribution? (Check all the statements that are true.)

The samples randomly generated from simulation cluster around 60.1 because that was the population mean assumed in the simulation. The samples randomly generated from bootstrapping cluster around 60.1 because that is the mean of the resampled data.

Let's say we took a small sample of A as well as a large sample of A, and calculated the mean of each sample. For which sample would our estimate have a wider 95% confidence interval?

The small sample of A.

If we calculate the favstats for our original sample of 157 thumb lengths (Fingers$Thumb); the bootstrapped sampling distribution of mean thumb lengths of samples of n=157 (bootSDoM$mean); and the simulated sampling distribution of means of samples of 157 thumb lengths (simSDoM$mean); which of these distributions would you expect to have similar means? Select the choices that would have similar means.

Thumb from Fingers Bmean from bootSDoM Cmean from simSDoM

𝑏0b0

Thumb length (in mm) when height is 0

𝑌𝑖

Thumb of each person (in mm)

# Estimate the standard error

Thumb.stats$sd / sqrt(157)

If short and tall populations do not really differ in thumb length, what would 𝛽1β1 be?

around 0

The estimate from our sample

b0 Thumb.stats$mean

Find the slope of the best-fitting regression line (i.e., calculate a value for 𝑏1b1)

b1(Thumb~Height...

If you have a fairly large sample, it won't really matter whether you use the t or z distribution as a model for your sampling distribution. If you have a small sample size, then this issue will matter a bit more. If you have a small sample size, which function will be a safer bet for you to use?

confint()

Given that the t distribution is more variable, and thus the margin of error will be slightly bigger to capture the middle .95 of sample means:

confint() result in a slightly bigger confidence interval than confint.default().

The lower bound of the confidence interval

confint(Empty.model)[1]

The upper bound of the confidence interval

confint(Empty.model)[2]

The margin of error in units of standard error

critical t xqt(.975, df = Thumb.stats$n - 1)

Which function could we add to repeat this sampling 1,000 times (to end up with 1,000 increments)?

do()

Repeat 1,000 times

do(1000)

95% of samples from our simulations have the same mean as our sample.

false

The t distribution is always the same, no matter the degrees of freedom.

false

The t distribution with small degrees of freedom is depicted in black.

false

There are only 95 samples that contradict our own sample in our simulations of 1,000 samples.

false

There is a 95% chance that our sample mean is 60.1 mm.

false

There is a 95% chance that we did our confidence interval calculation correctly.

false

t distribution

like the standard normal distribution, is a mathematical probability function that is a good model for sampling distributions of the mean. It's just a little more variable, that's all.

The lower bound for the confidence interval

lower bound of B0

Which of these distributions would you expect to have similar standard deviations? Select the choices that would have similar standard deviations.

mean from bootSDoM mean from simSDoM

What are the units of 58.73 and 61.48?

mm

Bootstrapping

resample from the actual data we have collected.

Resample with replacement to generate a new, bootstrapped sample

resample(Fingers, 157)

𝑒𝑖ei

residual (how differnt the actual thumb lengths is from the model's prediction for each person)

Standard error

s/sqrt(n) Thumb.stats$sd / sqrt(Thumb.stats$n)

# arrange simSDoM in descending order by mean

simSDoM <- arrange(simSDoM, desc(mean))

If we have 1,000 means in our sample, which of these represents the lower cutoff or the 25th mean from the bottom of the vector? (Check all that apply.) response - correct

simSDoM$mean[975] simSDoM$mean[1000-25]

Our sample would be considered likely to have come from any population with a mean within this confidence interval.

true

The normal distribution is always the same, no matter the degrees of freedom.

true

The normal distribution is depicted in black.

true

The t distribution is often more variable (that is, it has wider tails and is less clustered in the center) compared to the normal distribution.

true

The t distribution looks more like the normal distribution when we have more degrees of freedom to spend.

true

With a small number of degrees of freedom, the t distribution is more variable (that is, it has wider tails and is less clustered in the center) compared to the normal distribution.

true

The upper bound of the confidence interval

upper bound of B0

The blue lines in this image depict the boundaries for 95% likelihood. Would the boundaries for 99% boundaries be wider or narrower?

wider

Using this method, are we confident that the complex model is different from the simple model? response

yes

Which of the following shows you how to estimate standard error of the mean?

𝜎𝑌 = 𝜎/√n

We asked this before, but it bears repeating here. What would 𝛽1β1 have been if there was no real difference in thumb length in the population between short and tall people?

0 mm

Recall that the margin of error we found (from both simulation and bootstrapping) was about 1.4 mm. (see picture above). Which of the following are equivalent measures of this distance? (Check all that apply.)

2 standard deviations 2 standard errors 0.14 cm

Why are we interested in the lowest and highest 25 means?

25 is 2.5% of 1,000

If the 95% confidence interval is roughly 2 standard errors above and below the sample mean, how wide is the confidence interval?

4 standard errors across

Match the following labels to the corresponding part of the figure. Assume that the 1,000 means have already been arranged in descending order.

975th mean 225th mean 3Margin of error

Consider these abstract situations. Let's say there is a variable (called A) that is highly variable in the population. Let's say there is another variable (called B) that is less variable than A. If we estimate the mean of each variable, which estimate would have a wider 95% confidence interval?

A

Let's say we repeated our study and collected a new sample of 157 students' thumb lengths and a sample of one student's thumb length. Which of the following options is more likely?

This sample mean from 157 students will probably be more similar to our first sample mean (60.1) than the individual student's thumb length.

Let's test these ideas by applying it to the confidence interval we constructed in the previous section: 60.1 ±± 1.4 (you can read this as "60.1 plus or minus 1.4") Is 1.4 mm really equivalent to 2 standard errors? If so, how big would 1 standard error be?

About .7 mm

Given the sample mean we observed (our estimate), what is the range of possible values within which we could be 95% confident that the true population mean would lie?

Approximately 60.1 - 1.4 to 60.1 + 1.4 BApproximately 58.7 to 61.5 CThe sample mean plus or minus the margin of error DAll of the above

Which of these describes how to figure out this confidence interval using the normal approximation?

Assume that the cutoffs will be 2 standard errors above and below the sample mean; estimate the standard error using CLT; 𝑚𝑒𝑎𝑛±2 mean±2 * 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑒𝑟𝑟𝑜𝑟

If we assume that thumb length is normally distributed in the population, which formula would represent the range for 95% of individual thumb lengths?

𝜇 ± 2μ ± 2 * 𝜎

Why do you think the lower and upper cutoff points were approximately the same distance (~1.4) from the mean of the sampling distribution?

Because the sampling distribution of means is basically symmetrical, so the distance from the middle to the top 2.5% of means is the same as the distance from the middle to the bottom 2.5% of means.

Why would a small sample be more likely to produce a wider confidence interval than a larger sample? (Check all that apply.)

Because the standard deviation of the population divided by a smaller n would result in a bigger estimate of standard error. Because smaller samples are more variable, so the sampling distribution would be wider. Because the critical distance in terms of standard error (the t score) would be bigger with small samples.

Why is this increment different from 4.6?

Because this is the increment for a resampled set of data.

In a t distribution, which is more variable than the normal distribution, would we have to draw our boundaries farther out or closer in toward the mean to capture 95% of samples in the middle?

Farther out, because there are more sample means out in the tails so we have to go farther out to capture .95 of the means.

Let's say we collected just one new student's thumb length. What is the likelihood that this new individual score would be outside the 95% confidence interval of means?

Greater than 5%

𝑋𝑖

Height (in inches) of each person

R makes it easy to resample. For now, just know that there are various ways of getting a distribution of estimates. Most of the time they yield similar results. What is the key difference between simulation and bootstrapping? (Check all that apply.)

In bootstrapping, you assume the sampling distribution is only made up of individuals represented in your sample; in simulations, you don't. In simulation, you have to assume that the population will have a certain shape; in bootstrapping you don't.

Interpret the 5% where we are "not confident."

There is a 5% chance that our confidence interval does not include the true population mean.

What, exactly, do the five numbers printed in the R Console mean?

These are the means of new samples of 157 numbers resampled from Fingers.

What do you notice about all these critical t's?

They are all around 2 standard errors. They are closer to the critical z when degrees of freedom is larger. The critical t's are bigger when the sample size is smaller because the sampling distribution is more variable.

Take a look at these bootstrapped 𝑏1b1s. Which of the following statements about them are true? (Check all that apply.)

They vary. Some are smaller than 4.6 and some are larger than 4.6. The numbers should be interpreted in the same unit (mm) as the outcome variable (Thumb). They vary because they are random bootstrapped samples. They each represent a possible mean difference between the height groups.

Let's say we calculated 95% confidence intervals for the means of A and B, based on samples of n=100. Which confidence interval would be wider, measured in number of standard errors?

They would be equal in terms of their standard error, because both would be about four standard errors across. Using the standard error as a ruler adjusts for the variability in the population.

Does this figure prove that our sample came from a population with a 𝜇μ of 60.1?

This figure shows that a population with a mean of 60.1 could have generated our sample.

If we want to know whether there is a difference between height groups in the population, what are we actually trying to estimate

𝛽1

If the 95% confidence interval for pounds lost is -1.07 plus or minus .63, which of the following are true? (Check all that apply.)

We are 95% confident that the the mean of the DGP is in this range. If all housekeepers in the population participated in this program, we are 95% confident that the true mean number of pounds lost will be in this range.

Using the t distribution as a model of the sampling distribution of increments, are we pretty confident that the true population mean difference could be 0?

We are confident that it could be 0.

In bootstrapping a sampling distribution, what are we assuming?

We assume that all samples are randomly drawn from a population that is made up of the instances in our sample.

Note that we don't actually know 𝜎σ. What can we do in this case?

We can use the standard deviation of our sample, 𝑠s, as an estimate of sigma.

If we knew the size of the margin of error, how would we find the upper bound?

We would just add the margin of error to the sample mean.

If we knew the size of the margin of error, how would we find the lower bound?

We would subtract the margin of error from the sample mean.

Let's say the CI for mean thumb length is between 58.73 and 61.48. Would you expect the range that would capture 95% of individuals in the population to be wider or narrower than the CI?

Wider, because individuals vary more than means. To capture 95% of individuals we would have to go out farther.

Which parameter are you most interested in if you want to decide whether to reject the empty model and adopt the more complex one?

𝛽1

The second row is labeled "Height2Grouptall". What is estimated by this confidence interval (1.94, 7.26)?

𝛽1 The increment to add on to go from the Short to the Tall population.

Here is the model of the DGP we are trying to estimate: 𝑇ℎ𝑢𝑚𝑏𝑖=𝛽0+𝛽1𝐻𝑒𝑖𝑔ℎ𝑡2𝐺𝑟𝑜𝑢𝑝𝑖+𝜖𝑖Thumbi=β0+β1Height2Groupi+ϵi If there was no real difference in thumb length between short and tall people in the population, then we could simplify this model. Which of the following represent ways we can simplify our complex model? (Check all that apply.)

𝑇ℎ𝑢𝑚𝑏𝑖=𝛽0+0∗𝐻𝑒𝑖𝑔ℎ𝑡2𝐺𝑟𝑜𝑢𝑝𝑖+𝜖𝑖Thumbi=β0+0∗Height2Groupi+ϵi 𝑇ℎ𝑢𝑚𝑏𝑖=𝛽0+𝜖𝑖

Represents the best estimate of that relationship from our data

𝑌𝑖=𝑏0+𝑏1𝑋𝑖+𝑒𝑖

If this is the complex model we fit with our data (𝑌𝑖Yi = 𝑏0b0 + 𝑏1b1𝑋𝑖Xi + 𝑒𝑖ei), then how would we specify the empty model to compare it to?

𝑌𝑖=𝑏0+𝑒𝑖

Represents the relationship between Height and Thumb in the DGP

𝑌𝑖=𝛽0+𝛽1𝑋𝑖+𝜖𝑖

How would you find the margin of error approximated by the t distribution? (Check all that apply.)

𝑏0−𝛽0 b0−β0 lower bound 𝛽0β0 upper bound - 𝑏0b0 Critical t * standard error Critical t * sample standard deviation / 𝑛√ (𝛽0 β0 upper bound - 𝛽0β0 lower bound)/ 2

What would be the best estimate of this parameter? (Check all that apply.)

𝑏1 The sample difference of means

The first row in this output is labeled "(Intercept)". What is estimated by this confidence interval (55.94, 59.69)? (Check all that apply.)

𝛽0 The Short population's mean.