Sampling Population

Sampling Population

Hypothesis

An assumption that a researcher makes about some characteristic of the population under study.

Steps in Hypothesis Testing

Step One: Stating the Hypothesis

Null hypothesis: Ho

Alternative hypothesis: Ha

Step Two: Choosing the Appropriate Test Statistic

Hypothesis Testing

Step Three: Developing a Decision Rule

Step Four: Calculating the Value of the Test Statistic

• Use the appropriate formula
• Compare calculated value to the critical value.
• State the result in terms of:
• rejecting the null hypothesis
• failing to reject the null hypothesis

Step Five: Stating the Conclusion

Hypothesis Testing

Types of Errors in Hypothesis Testing

Type I Error

Rejection of the null hypothesis when, in fact, it is true.

Type II Error

Acceptance of the null hypothesis when, in fact, it is false.

Accepting Ho or Failing to Reject Ho?

One-Tailed Test or Two-Tailed Test?

Other issues

Type I and Type II Errors

Actual State of the Null Hypothesis

Fail to Reject Ho

Reject Ho

Ho is true

Ho is false

Correct (1-)

no error

Type II error ()

Type I error ()

Correct (1- )

no error

Independent Versus Related Samples

Independent samples

Measurement of a variable in one population has no effect on the measurement of the other variable

Related Samples

Measurement of a variable in one population may influence the measurement of the other variable.

Degrees of Freedom

The number of observations minus the number of constraints.

Commonly Used

Statistical Hypothesis Tests

Finding Confidence Intervals for Population Proportions

 = true population proportion (i.e., the parameter value)

Confidence Intervals for Population proportion:

p - 1.96sp    p + 1.96sp

p = proportion obtained from a single sample (i.e., the statistic value)

sp = estimate of the standard error of the sample proportion

p = number of sample units having a certain feature

total number of sample units (i.e., n)

sp =  { p (1-p) / n }

Finding Confidence Intervals for Population Proportions (Cont’d)

Given n = 100 and p = .64. To construct a 95 percent confidence interval for the population proportion

sp =

p (1 – p)

N

(.64)(.36) = .048

100

The 95 percent confidence interval is

p ± 1.96 sp = .64 ± (1.96)(.048)

= .64 ± .09408

= .64 ± .09, approximately.

Finding Confidence Intervals for Population Proportions (Cont’d)

• Interpretation
• This confidence interval can also be expressed in percentage terms: 64% ± 9%
• In other words, we can be 95 percent confident that between 55 and 73 percent of all general stores in the city carry boot polish

Factors Influencing Sample Size

• precision level
• confidence level
• variability
• Resources

Methods for Determining Sample Size

• The desired precision level
• The desired confidence level
• An estimate of the degree of variability in the population, expressed in the form of a standard deviation

Sample Size Estimation

zq2 s2

N = ------

H2

zqs

H = ----

n

• H-> Desired precision level
• q-> Desired confidence level
• S-> Sample Standard deviation
• N-> Population mean

Sample Size Estimation (Cont’d)

• A marketing manager of a frozen-foods firm wants to estimate within ±Rs10 the average annual amount that families in a certain city spend on frozen foods per year and have 99 percent confidence in the estimate
• He estimates that the standard deviation of annual family expenditures on frozen foods is about Rs100
• How many families must be chosen for this study?

Sample Size Estimation (Cont’d)

H = Rs10, s = Rs100, and zq = 2.575

(corresponding to a confidence level of 99 percent)

n = (2.575)2(100)2 = 663 families,approximately

(10)2

Determining Sample Size

• A sporting goods marketer wants to estimate the proportion of tennis players among high school students
• The marketer wants the estimate to be accurate within ±.02 and wants to have 95 percent confidence in the interval estimate
• A pilot telephone survey of 50 high school students showed that 20 of them played tennis. Estimate the required sample size for the final study from the given data

Determining Sample Size (Cont’d)

H = .02 and zq = 1.96. p = 20/50 =0.4

s = (20/50)(1 - 20/50) = (.4)(.6) = .24

z2q s2 (l.96)2(.24)2

n = ------------ = -----------------

H2 (.02)2

= 2,305 students, approximately

Advantages of Sampling

• Low Cost
• Reduced time

Thank You