AP Stat Ch 7-8

Question 0

Statistic vs parameter

Answer 0

Any quantity computer from the values in the sample is a statistic. X bar and s and p hat. These are variable

The corresponding population characteristic is usually called a parameter. Mu and sigma and p. These are constant

Question 1

Statistical inference

Answer 1

The process of drawing conclusions about a population based on data from a sample of that population

Question 2

Sampling variability

Answer 2

Since the value of the statistic (estimate) depends on the sample selected, the value of the statistic will vary from sample to sample. This variability is called sampling variability.
NOTE:
Sampling variability is the variability of X bar, not X
For example, if X bar = the average weight of a sample of 50 students, then X bar also has variability. This is sampling variability.

Question 3

Sampling distribution

Answer 3

The distribution of a statistic is called its sampling distribution. It is the distribution of all the values of the stat based on all possible samples of the same size.

Question 4

What to do when describing distribution

Answer 4

Say the shape center and spread!

Question 5

Bias

Answer 5

Sampling distributions allow us to describe bias more precisely by speaking of the bias of a statistic rather than bias in a sampling method. Bias concerns the center of the sampling distribution.

Question 6

Unbiased estimater

Answer 6

A statistic used to estimate a parameter is unbiased if the mean of its sampling distribution is equal to the true value of the parameter being estimated. The statistic is called an unbiased estimator of the parameter.

Question 7

Variability of a statistic

Answer 7

The variability of a statistic is described by the spread of its sampling distribution. This spread is determined by the sampling design and the size of the sample. Larger samples give smaller spread.

Question 8

Population proportion vs sample properition

Answer 8

Population is p and sample is p hat
P hat = X/n where X is the number of successes and n is the sample size.
X–> B(n,p)

Question 9

Mu p hat and sigma p hat

Answer 9

Mu p hat = p
Sigma P hat = SQRT (p(1-p)/n)
NOTE: population must be ten times as large as the sample to use this and p hat is only approx normal if np and n(1-p) are BOTH >=10

Question 10

Ex: suppose the proportion of HM students with a graphing calculator is p=.43. If you were to take a sample of size 50, what is the probability that in the sample, 30% or less had a graphing calculator?

Answer 10

Let p hat = sample proportion of people with a GC
Np and n(1-p) are both greater than or equal to 10 and assume the population is bigger than 500
So, mu p hat = .43 and sigma p hat = SQRT (.43(.57)/50) = .07
Z = (.3-.43)/.07=-1.86. Norm cdf (-100000000,-1.86,0,1)
If sample size is only ten, then can’t use normal approx. X–> B (10,.43) and use that

Question 11

Sample mean rules:

Answer 11

1. Mu sub x bar = mu
2. Sigma sub x bar = sigma / SQRT (n)
3. Pop needs to be at least ten times the sample
4. n needs to be greater than 30– CLT

Question 12

Ex. Suppose that the population of residents of a local town has a mean annual income of 50000 and standard deviation of 35000. If a random sample of 50 residents is selected, what is the probability that their sample mean is within $5000 of the true mean?

Answer 12

n is >= 30 and assume 500 people in the town
So, mu sub x bar= 50000 and sigma x bar = 35000/SQRT(50)
X = 45000-50000 / sigma sub x bar
Then use norm cdf

Question 13

What influences variability?

Answer 13

SAMPLE SIZE, NOT POPULATION SIZE

Question 14

Point estimate

Answer 14

A single number that represents a plausible value for the population characteristic. It is a single point on the number line. For example, we estimate that61% of adults favor…

Question 15

Interval estimate

Answer 15

A range of plausible values for that characteristic. Interval on a number line.

Question 16

Two things to look for when choosing a statistic to estimate a population characteristic:

Answer 16

No bias and low variability.

Question 17

Unbiased statistic

Answer 17

Statistic with a mean value = to that of the population characteristic being estimated. Centered in the right place

Question 18

Commonly used stats:

Answer 18

X bar for mu
P hatfor p
s for sigma

Mean, proportion, standard deviation

Question 19

Confidence interval

Answer 19

Point estimate not the only plausible value, so statisticians will usually report an interval of plausible values for the population characteristic based on the sample. This interval is called a confidence interval.
More likely to be correct than a point estimate!

Question 20

General form of a confidence interval:

Answer 20

Point estimate +- (multiplier) (standard error)

Question 21

Confidence interval for a parameter has two parts:

Answer 21

1. An interval calculated from the data with form estimate +- margin of error.
2. Confidence level, c, which gives the overall success rate of the method for calculating the confidence interval.

Question 22

What does confidence level of 95% mean?

Answer 22

95% of all intervals calculated in same way should incorporate the true population mean

Question 23

Standard error of p hat

Answer 23

SE = SQRT ((p hat)(1-p hat) / n)

Question 24

Calculating one sample z confidence interval for a population proportion

Answer 24

P hat +- z STAR * SQRT ((p hat)(1-p hat)/n)

Question 25

How to find z star

Answer 25

For confidence level C,

Z star = Abs (inv norm (1-c)/2)

Question 26

One sample z confidence interval for a population proportion 4 step plan

Answer 26

1. Name of test– 1 sample, z, 95% confidence interval for p = true proportion of…
2. A. Random sample?
   B. Large sample size – n (p hat) >=10 and n(1-p hat)>10
   C. Independence – N >= 10n
3. Calculation:
   Interval = P hat +- Z* * SQRT (p hat(1-p hat)/n)
4. Conclusion: I am c% confidence that the true proportion of … is between the interval… and …

Question 27

Margin of error

Answer 27

The plus minus part in the confidence interval

ME = z* * SQRT (p hat(1-p hat)/n)

Question 28

How to calculate sample size needed when given a desired margin of error

Answer 28

Assume p hat =.5

Then plug into the equation ME = z star * SQRT (p hat(1-p hat)/n) and solve for n. Round up!

Question 29

Benefit and tradeoff of a bigger margin of error

Answer 29

More likely that p is in the interval

But, much less precision in a wider interval

Question 30

Benefit and tradeoff of decreasing ME

Answer 30

Increase sample size –> more precision

But it takes longer, more expensive

Question 31

When to use a z interval for a population mean

Answer 31

Only when sigma, the population standard deviation is known!
Need to meet the conditions:
1. SRS
2. Normality 
3. Independence (N>=10n)

Question 32

If we don’t know sigma for an interval with the population mean, what do you do?

Answer 32

Use s and do a t interval instead of sigma and a z interval

Standard error = s/SQRT (n)

Question 33

Sample size for a desired margin of error with means

Answer 33

ME >= z (sigma)/SQRT (n)

Question 34

T distribution

Answer 34

Use for a mean when the population standard deviation is unknown.
Not a normal curve– the ends have more area because more variability because depends on the mean and standard deviation, two statistics.

Question 35

Degrees of freedom

Answer 35

n -1 where n is the sample size

Question 36

What happens as the degrees of freedom increase?

Answer 36

More approximately normal

Larger sample, better we are. As n increases, s –> sigma and the t distribution –> z distribution (normal curve)

Question 37

4 step procedure for a t interval:

Answer 37

1. C% t interval for mu = the true mean of …
2. A. Random sample?
   B. Sample size is large (n>=30) or the population is normal (graph the data, check and discuss the shape. If outliers or extremely skewed, then procede with caution)
   C. N >= 10n
3. Calculate the interval–> x bar +- t* (s / SQRT (n))
4. Interpret– C% confident true mean of … is from ___ to ___

Question 38

Paired t procedures

Answer 38

Same as normal t procedures, but the mean is the average difference in the responses to the two treatments within matched pairs of subjects in the entire population or the mean difference in response to the two treatments for individuals in the population (when the same subject receives both treatments) or the mean difference between before and after measurements for all individuals in the population.

Question 39

Sample size for t procedure

Answer 39

If n<15, then needs to be close to normal. If outliers or clearly not normal, don’t use t procedures
If n >= 15, then t can be used except with outliers or strong skewness
If n>=30, then can use no matter what.