Population All members of a set which have a given characteristic. Population Data Data associated...

•Population All members of a set which have a given characteristic.•Population Data Data associated with a certain population.•Population Parameter A measure that is computed using population data.•Sample A subset of the population.

7.1 Definitions

•Statistic An estimated population parameter computed from the data in the sample.•Random Sample A sample of which every member of the population has an equal chance of being a part.•Bias A study that favors certain outcomes.•Point Estimate A single number (point) that attempts to estimate an unknown parameter.

Population(Size of population = N)

Sample number 1

Sample number 2

Sample number 3

Sample number NCn

Each sample size = n

Example:Let N = {2,6,10,11,21} Find µ, median and σ

µ = 10median = 10 σ = 6.36

How many samples of size 3 are possible? n ave med Sx σ

1 2,6,10 6 6 4 3.26

2 2,6,11 6.33 6 4.5 3.68

3 2,6,21 9.67 6 10.01 8.17

4 2,10,11 7.67 10 4.93 4.03

5 2,10,21 11 10 9.53 7.79

6 2,11,21 11.33 11 9.50 7.76

7 6,10,11 9 10 2.64 2.16

8 6,10,21 12.33 10 7.77 6.34

9 6,11,21 12.67 11 7.63 6.24

10 10,11,21 14 11 6.08 4.97

5C3 = 10

n

xμ ,x of average x

10μx

med 6 10 11P(x) .3 .4 .3

Point Estimators

p

s

x

N n

Sample Population

N

xμ

N

1ii

n

xX

N

1ii

N

μ)(xσ

N

1i

2i

1-n

)X(xs

n

1i

2i

Sample Population

1 092725 012157 827052 297980 625608 9641342 104460 007903 484595 868313 274221 3671813 676071 388003 266711 323324 044463 7628034 881878 862385 203886 261061 096674 8115485 534500 336348 086585 241740 581286 0084356 094276 615776 242112 985859 075388 0820037 333848 513630 474798 841425 331001 5427408 847886 629263 596457 589243 576797 8009579 942495 695172 523982 264961 771016 11879710 450553 679145 324036 715835 963418 53304811 024670 615375 717260 171144 340939 20871212 932959 205554 113225 704406 263818 633643

Random Number Table

Characteristics of a Sampling Distribution

Definition: The sampling distribution of the X’s is a frequency curve, or histogram constructed from all the NCn

possible values of X.

μμx

Characteristic 2. The standard deviation of the sampling distribution which is the standard deviation of all the NCn of X is equal to the standard deviation of the population divided by the square root of the sample size. (Also called the standard error, SE.)

n

σσx

Characteristic 1. The mean of all the NCn possible values of X is equal to the population mean, µ.

Assumptions

n > 30The sample must have more than 30 values.

Simple Random SampleAll samples of the same size have an equal

chance of being selected.

Large Samples

Definitions Estimator

a formula or process for using sample data to estimate a population parameter Estimate

a specific value or range of values used to approximate some population parameter Point Estimate

a single value (or point) used to approximate a population parameter

The sample mean x is the best point estimate of the population mean µ.

The sample standard deviation s is the best point estimate of the population standard deviation .

The sample proportion p is the best point estimate of the population proportion .

Definitions

Confidence Interval

(or Interval Estimate)

A C% confidence interval for a population mean, μ, is an interval [a,b] such that μ would lie within C% of such intervals if repeated samples of the same size were formed and interval estimates made.

Central Limit Theorem

Under certain conditions, the sampling distribution of the X’s result in a normal distribution

Definition

Confidence Interval


a range (or an interval) of values used to estimate the true value of the population

parameter

Lower # < population parameter < Upper #

Confidence Interval


a range (or an interval) of values used to estimate the true value of the population

parameter

Lower # < population parameter < Upper #

As an example

Lower # < < Upper #

the probability 1 - (often expressed as the equivalent percentage value) that is the relative frequency of times the confidence interval actually does contain the population parameter, assuming that the estimation process is repeated a large number of times

usually 90%, 95%, or 99% ( = 10%), ( = 5%), ( = 1%)

Degree of Confidence(level of confidence or confidence coefficient)

Interpreting a Confidence Interval

Correct: We are 95% confident that the interval from 98.08 to 98.32 actually does contain the true value of

. This means that if we were to select many different samples of size 106 and construct the confidence intervals, 95% of them would actually contain the

value of the population mean .

Wrong: There is a 95% chance that the true value of will fall between 98.08 and 98.32.

98.08o < µ < 98.32o

Confidence Intervals from 20 Different Samples

the number on the borderline separating sample statistics that are likely to occur from those that

are unlikely to occur. The number z/2 is a critical

value that is a z score with the property that it separates an area /2 in the right tail of the standard normal distribution.

Critical Value

The Critical Value

z=0Found from Table A-2

(corresponds to area of

0.5 - 2 )

z2

z2-z2

2 2

Finding z2 for 95% Degree of Confidence

-z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%


-z2z2

95%

.95

.025.025

2 = 2.5% = .025 = 5%

Critical Values


.4750

.025

Use Table A-2 to find a z score of 1.96

= 0.025 = 0.05



.025.025

- 1.96 1.96

z2 = 1.96

.4750

.025

Use Table A-2 to find a z score of 1.96

= 0.025 = 0.05

Margin of ErrorDefinition

Margin of Error is the maximum likely difference observed between sample mean x and true population

mean µ.

denoted by E

Definition


mean µ.

denoted by E

µ x + Ex - E

Definition


mean µ.

denoted by E

µ x + Ex - E

x -E < µ < x +E

Definition


mean µ.

denoted by E

µ x + Ex - E

x -E < µ < x +Elower limit

Definition

upper limit

Definition Margin of Error

µ x + Ex - E

E = z/2 •n

Definition Margin of Error

µ x + Ex - E

also called the maximum error of the estimate

E = z/2 •n

Calculating E When Is Unknown

If n > 30, we can replace in Formula 6-1 by the sample standard deviation s.

If n 30, the population must have a normal distribution and we must know to use Formula 6-1.

x - E < µ < x + E

(x + E, x - E)

µ = x + E

Confidence Interval (or Interval Estimate)

for Population Mean µ(Based on Large Samples: n >30)

Procedure for Constructing a Confidence Interval for µ

( Based on a Large Sample: n > 30 )

1. Find the critical value z2 that corresponds to the desired degree of confidence.

3. Find the values of x - E and x + E. Substitute thosevalues in the general format of the confidenceinterval:

4. Round using the confidence intervals roundoff rules.

x - E < µ < x + E

2. Evaluate the margin of error E = z2 • / n . If the population standard deviation is unknown, use the value of the sample standard deviation s provided that n > 30.

Example: A study found the body temperatures of 106 healthy adults. The sample mean was 98.2 degrees and the sample standard deviation was 0.62 degrees. Find the margin of error E and the 95% confidence interval.

n = 106

x = 98.20o

s = 0.62o

= 0.05/2 = 0.025

z / 2 = 1.96

nE = z / 2 • = 1.96 • 0.62 = 0.12

106


x - E < < x + E

98.20o - 0.12 < < 98.20o + 0.12

98.08o < < 98.32o


Based on the sample provided, the confidence interval for the population mean

is 98.08o < < 98.32o. If we were to select many different samples of the same size,

95% of the confidence intervals would actually

contain the population mean .

Finding the Point Estimate and E from a Confidence Interval

Point estimate of µ:

x = (upper confidence interval limit) + (lower confidence interval limit)

2

Margin of Error:

E = (upper confidence interval limit) - (lower confidence interval limit)

2

Population All members of a set which have a given characteristic. Population Data Data associated...

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