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 N C n Each sample size = n

Example: Let N = {2,6,10,11,21} Find µ, median and σ µ = 10 median = 10 σ = 6.36 How many samples of size 3 are possible? navemedSxSx σ 1 2,6, ,6, ,6, ,10, ,10, ,11, ,10, ,10, ,11, ,11, C 3 = 10 med P(x).3.4.3

Point Estimators Sample Population

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 N C n possible values of X. Characteristic 2. The standard deviation of the sampling distribution which is the standard deviation of all the N C n 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.) Characteristic 1. The mean of all the N C n possible values of X is equal to the population mean, µ.

Assumptions  n > 30 The sample must have more than 30 values.  Simple Random Sample All 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 (or Interval Estimate) a range (or an interval) of values used to estimate the true value of the population parameter Lower # < population parameter < Upper #

Confidence Interval (or Interval Estimate) 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 to 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 and o < µ < o

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=0 Found from Table A-2 (corresponds to area of   2 ) z  2 -z  2  2

Finding z  2 for 95% Degree of Confidence

-z  2 z  2 95%  2 = 2.5% =.025  = 5% Finding z  2 for 95% Degree of Confidence

-z  2 z  2 95%  2 = 2.5% =.025  = 5% Critical Values Finding z  2 for 95% Degree of Confidence

Use Table A-2 to find a z score of 1.96  =  = 0.05 Finding z  2 for 95% Degree of Confidence

z  2 = 1.96   Use Table A-2 to find a z score of 1.96  =  = 0.05

Margin of Error Definition

Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E Definition

Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E Definition

Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E x -E < µ < x +E Definition

Margin of Error is the maximum likely difference observed between sample mean x and true population mean µ. denoted by E µ x + E x - E x -E < µ < x +E lower limit Definition upper limit

Definition Margin of Error µ x + E x - E E = z  /2  n

Definition Margin of Error µ x + E x - 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 z  2 that corresponds to the desired degree of confidence. 3. Find the values of x - E and x + E. Substitute those values in the general format of the confidence interval: 4. Round using the confidence intervals roundoff rules. x - E < µ < x + E 2. Evaluate the margin of error E = z  2  / 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 = o s = 0.62 o  = 0.05  / 2 = z  / 2 = 1.96 n E = z  / 2  = = 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. x - E <  < x + E o <  < o o <  < o

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. Based on the sample provided, the confidence interval for the population mean is o <  < o. 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