Chapter 6-8 Test Prep

Question 1

Backing into σ for confidence intervals

Answer 1

if you assume a normal distribution, the range is approximately equal to 6σ (± 3σ) so that you estimate σ as the range divided by 6

Question 2

Central Limit Theorem

Answer 2

As the sample size gets large enough, the sampling distribution of the mean is approximately normally distributed. This is true regardless of the shape of the distribution of hte individual values in the population.

Question 3

Confidence Interval Estimate for the Proportion

Answer 3

where

p = sample proportion = X/n

π = population proportion

Z_α/2 = critical value from the standardized normal distribution

n = sample size

Question 4

Confidence Interval Estimate of the Mean

Answer 4

(1- α)*100%

99% confidence interval would be

(1 - α)*100% = 99% > (1 - α) = .99 > 1 - .01 = α > α = .01

Find 1 - α/2 to get the area under the curve for 99%

then find Z_α/2 = Z_0.005 >

Question 5

Confidence Interval for the Mean (σ unknown)

Answer 5

where

t_α/2 is the critical value corresponding to an upper-tail probability of α/2 from the t distribution with n - 1 degrees of freedom

Question 6

Confidence Interval for the Mean (σ known)

Answer 6

This is essentially a “known probability” (which is for a population) but for a sample.

i.e. you’re substituting x̄ for µ

Question 7

cumulative standardized normal distribution

Answer 7

Represents area under the cumulative standardized normal distribution from -∞ to Z

Question 8

D.C.O.V.A.

Answer 8

Define the data that you want to study to solve a problem or meet an objective

Collect the data from appropriate sources

Organize the data collected, by developing tables

Visualize the data collected, by devleoping charts

Analyze the data collected, to reach conclusions and present those results

Question 9

Degrees of Freedom

Answer 9

n - 1

Question 10

exponential distribution

Answer 10

Contains values from zero to positive infinity and is right-skewed, making the mean greater than the median

Question 11

Finding an X Value Associated w/ a Known Probability

Answer 11

X = µ + Zσ

Question 12

Finding Z for the Sampling Distribution of the Mean

Answer 12

Finding how a sample mean, x̅, differs from the population mean. By substituting x̅ for x, µ_x̅ for µ, and σ_x̅ for σ in the equation for finding Z, you get the following:

Question 13

Finding Z for the Sampling Distribution of the Proportion

Answer 13

Question 14

Finding x̄ for the Sampling Distribution of the Mean

Answer 14

To find the interval that contains a specific proportion of the sample means.

Note:

Instead of X = µ + σ (i.e. the standard deviation) you use X̅ = µ + Z*σ/sqrt-n (standard error of the mean)

Question 15

Level of Confidence

Answer 15

is symbolized by (1 - α) * 100%, where

α is the proportion in the tails of the distribution that is outside the confidence interval

Question 16

Normal Probability Density Function

Answer 16

A probability density function for a specific continuous probability distribution defines the distribution of the values for a continuous variable and can be used as the basis for calculations that determine the likelihood or probability that a value will be within a certian range

Question 17

Population Mean

Answer 17

Question 18

Population Standard Deviation

Answer 18

The Standard Deviation is a measure of how spread out numbers are.

Question 19

Sample Proportion

Answer 19

It is the proportion of items in the sample with the characteristics of interest

The sample proportion, p, will be b/t 0 & 1.

Question 20

Sample Size Determination for the Mean

Answer 20

To compute the Sample Size, you must know three quantities:

1. The desired confidence level, which determines the value of Zα/2, the critical value from the standardized normal distribution
2. The acceptable sampling error, e
3. The standard deviation, σ

Question 21

sampling distribution

Answer 21

A sampling distribution is the distribution of the results if you actually seloected all possible samples.

Question 22

sampling distribution of the mean

Answer 22

the distribution of all possible sample means if you select all possible samples of a given size

Question 23

sampling distribution of the proportion

Answer 23

the distribution of all possible sample means if you select all possible sample proportions of a given size

However, you can use the normal distribution to approximate the binomial distribution when

nπ and n(1 - π) are each at least 5

Question 24

Sampling error e

Question 25

Standard Confidence Intervals

90%

95%

99%

Answer 25

90% = .95 & .05

95% = .975 & .025

99% = .995 & .005

Question 26

Standard Error of the Mean

Answer 26

The value of the standard deviation of all possible sample means, called the standard error of the mean, expresses how the sample means vary from sample to sample.

Question 27

Standard Error of the Proportion

Answer 27

The statistic p is an unbiased estimator of the population proportion, π

By analogy to the sampling distribution of the mean, whose standard error is σ_X̅ = σ/sqrt-n, the

STANDARD ERROR OF THE PROPORTION IS

Question 28

Student’s t distribution

Answer 28

t = x̄ - µ / (S / sqrt-n)

If the variable X is normally distributed, the the following statistic has a t distribution with n - 1 degrees of freedom

Question 29

The Unbiased Property of the Sample Mean

Answer 29

The sample mean is unbiased because the mean of all the possible sample means (of a given sample size, n) is equal to the population mean, µ.

Question 30

Theoretical Properties of Normal Distribution

Answer 30

• Symmetrical Distribution - therefore the mean and median are equal
• Bell shaped. Values cluster around the mean
• Interquartile range is roughly 1.33 σ. Therefore, the middle 50% of the values are contained within an interval that is approximately 2/3’s of a standard deviation below and above the mean
• the distribution has an infinite range. Six σ approximate this range.

Question 31

uniform distribution

Answer 31

Contains values that are equally distributed in the range between the smallest value and the largest value. (aka rectangular distribution)

Question 32

Values for π and π(1-π)

Answer 32

When π = 0.9, then π(1-π) = (0.9)(0.1) = 0.09

When π = 0.7, then π(1-π) = (0.7)(0.3) = 0.21

When π = 0.5, then π(1-π) = (0.5)(0.5) = 0.25

When π = 0.3, then π(1-π) = (0.3)(0.7) = 0.21

When π = 0.1, then π(1-π) = (0.1)(0.9) = 0.09

Question 33

Z Transformation Formula

Answer 33

The transformation formula computes a Z value that expresses the difference of the X value from the mean, µ, in standard deviation units called standardized units.

Then you can determine the probabilities by using the cumulative standardized normal distribution table.

The Z value is equal to the difference between X and the mean, µ, divided by the standard deviation, σ.

Question 34

What 3 things must you know to determine a sample size of a proportion

Answer 34

1. The desired confidence lev el, which determines the value of Z_α/2, the critical value from the standardized normal distribution
2. The acceptable sampling error (or margin of error), e
3. The population proportion, π

Question 35

Movement of standard error

Answer 35

As the standard error decreases, values become more concentrated around the mean.

Therefor, the probability that the sample mean will fall in a region that includes the population mean will always increase when the sample size increases.