STSCI 2150 Prelim #3 Review

The normal distribution is (continuous/discrete).

continuous

True/False: In a normal distribution, the mean, median, and mode are the same.

True.

What is the mean and standard deviation of the standard normal distribution?

The mean is 0 and the standard deviation is 1.

In a normal distribution, 95% of the values are within __________ standard deviations of the mean.

1.96

What is a Z-score?

A Z-score is the number of standard deviations from the mean.

What is Z-transformation?

When you are answering a question about a given normal distribution, you can transform it into a standard normal distribution by finding the Z-score and then using 0 for the mean and 1 for the standard deviation.

How do you calculate the Z-score?

(score-mean)/SD

True/False: Z-scores cannot be negative.

False. A value under the mean will have a negative Z-score.

The normal distribution statistical tables give the probability density (above/below) the given value.

above

The pnorm() function gives the probability density (above/below) the given value.

below

How do you find pr(Z > 1.73)? Describe methods for using R and a statistical table.

Using R: 1 - pnorm(1.73)

Using tables: Read the table (don't need to subtract from 1).

How do you find pr(Z < -1.89)? Describe methods for using R and a statistical table.

Z > -1.89 is the same as Z > 1.89.

Using R: 1 - pnorm(1.89)

Using tables: Read the table for 1.89 (don't need to subtract from 1).

How do you find pr(Z < 1.61)? Describe methods for using R and a statistical table.

Using R: pnorm(1.61)

Using tables: 1 - table reading

In the U.S., female heights are normally distributed with a mean of 64.3 inches and an sd of 3.9 inches. A woman is 74 inches tall. What is her percentile? (Use R and a table)

Using R: pnorm(74, mean=64.3, sd=3.9) = 0.994.

Using tables: (74 - 64.3)/3.9 = 2.49 = Z-score. Read the table to find 0.00639 (the area under the curve above 74 inches). 1 - 0.00639 = 0.994.

The normal distribution statistical table yields the area under the curve for a value (above/below) the given value.

above

The pnorm() function yields the area under the curve for a value (above/below) the given value.

below

True/False: The statistical tables directly give you the percentiles for the normal distribution.

False (pnorm() gives the percentile).

How can you use normal distribution statistical tables to find the value corresponding to a given percentile?

Look at the table to find the percent corresponding to the percentile. The negative of the Z-score corresponding to that number is the Z-score for the percentile. You can plug this Z-score into (X - mean)/sd to solve for X.

True/False: The normal distribution statistical tables only show values for negative Z-scores.

False.

How do you use normal distribution statistical tables to determine the value at the 98.5 percentile?

Find 0.015 on the table and the corresponding Z score (because the statistical table represents the area to the right).

How do you use normal distribution statistical tables to determine the value at the 4th percentile?

Since this value is below the 50th percentile, the Z-score is negative. The value under a negative Z-score is equal to the value above the positive version of the Z-score, so find 0.04 on the statistical table, and multiply the Z-score by negative 1.

What is the R code to be used for finding out how tall a man is at the 30th percentile (where the population mean is 177 and the standard deviation 7.1)?

qnorm(0.3, 177, 7.1)

What does the qnorm() function output?

qnorm() outputs the value at a particular percentile.

What does the pnorm() function output?

pnorm() outputs the percentile of a given value.

This is referring to a normal distribution where the population mean is 177 and the standard deviation is 7.1. Write the R code.

What percentage of the population is taller than 175?

1 - pnorm(175, 177, 7.1)

In the pnorm function, do you input the value or the Z-score?

value

This is referring to a normal distribution where the population mean is 177 and the standard deviation is 7.1. Write the R code.

What is the 80th percentile value of this population?

qnorm(0.8, 177, 7.1)

This is referring to a normal distribution where the population mean is 177 and standard deviation is 7.1. Write the R code.

What is the 40th percentile value of this population?

qnorm(0.4, 177, 7.1)

True/False: Z scores are only for use with statistical tables and not for R.

True.

The Z-score of a given value is -0.28. The value in the statistical table corresponding to the Z-score of 0.28 is 0.38974.

What proportion of the population is to the right of -0.28?

0.38974 corresponds to the probability density below -0.28. Thus, we need to do 1 - 0.38974, which is 0.61026.

What does the dnorm() function output in R?

dnorm() outputs the density for a particular value on a normal curve (the corresponding y-value for a dot on the curve).

What does the rnorm() function output in R?

rnorm() randomly generates normally distributed values from a normal distribution.

rnorm() and dnorm() deal with the (points on a normal curve/area under a normal curve).

points on a normal curve

pnorm() and qnorm() deal with the (points on a normal curve/area under a normal curve).

area under a normal curve

Means of random samples of size n are drawn from a normally distributed population with a given mean and standard deviation. Describe the sampling distribution of the means (shape, center, spread).

shape = normal
center = mean
spread = sd/√n

What is the formula for standard error?

sd/√n

True/False: The area under a sampling distribution (of samples from a normally distributed population) will be less than 1.

False (all normal curves have an area under the curve of one!).

For normal curves, how much taller is the peak of a sampling distribution compared to the original curve?

The number of times smaller the standard deviation is than the original curve sd is the factor by which the sampling distribution curve is taller.

If we have data that we know to be coming from a normal distribution with 𝜇 = 200, 𝜎 = 10, what would the standard error be for samples sizes of 4?

5

For normal sample distributions, as sample size increases, the distribution of sample means gets (wider/narrower).

narrower

The greater the sample size, the (larger/smaller) the standard error.

smaller (standard error is sd/√n, so an increased sample size would make the value smaller)

A given population distribution is normal with a mean of 80 and a standard deviation of 20. Sketch population and sampling distribution for samples with the size of 100.

Sketch.

What is the Central Limit Theorem?

Regardless of the population distribution, the sampling distribution will attain a more normal distribution as the sample sizes increase.

True/False: A population has to be normally distributed in order for the sampling distribution to be normal.

False (according to the central limit theorem, the mean of a large number of measurements, randomly sampled from any type of distribution, is approximately normally distributed).

See the three distributions on the 'Prelim 3 Things to Remember' document. Rank (a), (b), and (c) in order of increasing sample size.

b, a, c (because the sampling distributions get more normally distributed as the sample sizes increase)

What are three ways to make an inference?

(a) Use a point estimate (i.e. to make an inference about the population mean, take the mean of a sample), (b) make a confidence interval, (c) do a hypothesis test.

The original formula for a confidence interval is (Y bar - 1.96(sd), Y bar + 1.96(sd)). What is the formula for an inference?

(Y bar - t(alpha)(2),df*(SE of Y bar), Y bar + t(alpha)(2),df*(SE of Y bar))

Note that we are not using the population standard deviation at all!

True/False: When calculating a confidence interval using the t-distribution, use the population standard deviation for the standard error calculations.

False (Use the mean and standard deviation from the sample)

What is the notation for the quantile for the t-distribution?

t alpha (1 or 2, depending if it's 1 or 2 sided), df

What does degrees of freedom mean?

The degrees of freedom is the maximum number of logically independent values.

The population mean for a distribution is 6.8 and sd is 2.1. A sample of 6 is taken from this population, and Y bar is 6.125 and s is 1.394. Calculate the 95% confidence interval for the mean.

(Y bar - (t quantile)(SE Y bar), Y bar + (t quantile)(SE Y bar)

(4.66, 7.59)

Why is it better to use a t-distribution when making an inference?

(Y bar - population mean)/(s√n) has a t distribution.

What are two ways to calculate the t quantile?

(1) Using statistical tables.

(2) Using R: qt(1 - alpha/2), df)

What are the inputs and outputs of qt()?

Input: 1 - alpha/2, df

Output: t quantile

True/False: To get the t-quantile from qt(), put in 1 - alpha and df.

False (1 - alpha/2)

What are two reasons why a 95% confidence interval for a sample of 6 could be relatively wide?

(1) Small sample
(2) Large sd

The student's t distribution has fatter tails than the normal distribution. What does this mean?

The curve for the t has a lower peak and higher tails.

The peak of the curve of the t distribution is (higher/lower) than the curve of the normal distribution.

lower

The ends of the t distribution are (higher/lower) than the tails of the normal distribution.

higher

What does the t-quantile represent on a curve?

The areas under the curve on their side of the curve are equal to alpha/2 (in total, the rea is alpha). The t-quantile and -t-quantile are the boundaries on the x axis for these areas.

As the number of degrees of freedom decreases, the tails of the t-distribution get (lighter and thinner/fatter and heavier).

fatter and heavier

As the number of degrees of freedom decreases, the peak of the t-distribution gets (higher/lower).

lower

The confidence interval gets (wider/narrower) as sample size increases.

narrower

As sample sizes increase, the t-distribution gets (more/less) normal.

more

As the degrees of freedom increase, the confidence interval gets (wider/narrower).

narrower (corresponds to a greater sample size)

As the alpha value increases, the confidence interval gets (wider/narrower).

narrower

For a sample size greater than ______, we can use the percentiles of a normal distribution for the confidence interval of a population mean (i.e. 1.96) instead of the t-distribution.

200

Why is the 2SE rule of thumb worse than the t-distribution method for determining a 95% confidence interval?

The 2SE method assumes a normal distribution, which only happens at very high sample sizes. Smaller samples have a t-distribution. Also, because the 2SE rule uses 1.96 (the Z-score for 95%), it's only good for 95% confidence intervals.

What is an assumption used for the confidence interval of the population mean?

The population is normally distributed

A study is comparing photosynthesis rates on 10 randomly chosen spruces compared to the rate measured on a red cedar growing next to each spruce. Is this a paired or two-sample t-test?

paired

A study is comparing photosynthesis rates for 10 randomly-chosen Douglas-fir trees compared to 10 randomly chosen red cedars. Is this a paired or two-sample t-test?

two-sample

What is the parameter for a paired t-test (i.e. what do you use for the null and alternative hypotheses)? The point estimate? The test statistic?

μd is the parameter (the mean of the differences), d bar is the point estimate (the average difference), and the test statistic is tobs

What is the null hypothesis in a paired t-test?

μd = 0

What is the test statistic for a paired t-test?

tobs (d bar/(s/√n))

Go through the steps of TUNA TEA with a paired t-test.

TU = think up a research question

N = H0: μd = 0

A = HA: μd =/= 0

T = tobs (d bar/(s√n))

E = (1) COMPARE TO TEST STATISTIC. t(alpha)(2)(df). If this critical value is less than the test statistic, you can reject the null hypothesis. (2) CALCULATE THE P-VALUE. 2*(1 - pt(tobs, df))

A = assess

The determinant of the spread of the t-distribution is ______.

df

How do you calculate the degrees of freedom in a paired t-test?

number of pairs - 1

When creating a confidence interval using the t method, how does the calculation change depending on the confidence interval (i.e. what in the calculations makes a 99% confidence interval wider than a 95% confidence interval?)?

the t quantile

What do you have to input into the qt() function? What is the output?

Input: (1-alpha/2), df
Output: t quantile

What is the R code used to determine the p-value of a two sided paired t-test? (List all 3 ways)

(1) 2*(1-pt(tobs, df))
(2) pt(-tobs, df) + (1-pt(tobs, df))
(3) 2*pt(-tobs, df)

What are the two assumptions of paired t-tests and confidence intervals for μd?

(1) The sample is a random sample of pairs, and (2) the distributions of the differences has to be normal.

True/False: To conduct a paired t-test or calculate a confidence interval, one of the assumptions is that the distribution of values for each side of the pair is normal.

False(ish). The distribution of the differences has to be approximately normal.

In a paired t-test, if the distribution of the differences is approximately normal with one or two outliers far from the normal curve, are the assumptions fulfilled?

No

How do you calculate the confidence interval for a paired t-test?

d bar +/- t*df

True/False: Paired designs have more power. Why or why not?

True because they minimize confounding variables.

What are the three types of inferences?

(1) point estimate, (2) hypothesis test, (3) confidence interval

What are the three estimates for a paired t-test?

(1) d bar, (2) t-test, (3) CI

What distribution(s) involved in a two-sample t-test is assumed to be normally distributed?

distribution of each group (separately)

What distribution(s) involved in a paired t-test is assumed to be normally distributed?

distribution of the differences

What is the general form of a confidence interval?

point estimate +/- t(alpha)(2),df(SE)

True/False: tobs (the test statistic) is used in the calculation of confidence intervals.

False.

How do you calculate the degrees of freedom for a two-sample t-test?

n1 + n2 - 2

What is the null value for a two-sample t-test?

0

What is sp^2, which is used in the calculation of the confidence interval in a two-sample t-test?

pooled variance (weighted average of the variance in each of the groups)

What is the null value for a two sample t-test?

0

Does order of the groups (i.e. which is subtracted from which) make a difference in the conclusion of a two-sample or paired t-test?

No

What is the general formula for a t-test test statistic?

tobs = point estimate/SE

How do you determine if a tobs is the right value for rejecting the null hypothesis?

|tobs| > critical value = reject null

True/False: We consider |tobs| when evaluating extremeness by comparing to a critical value.

True.

What are the three assumptions for the two-sample t-tests?

(1) Random sample for each population, (2) normal distribution of measurements in each population group, (3) approximate same variances in each group