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Sampling Distribution and Hypothesis Testing

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Chapter 10 : Sampling Distribution and Hypothesis Testing

Sampling Distribution and Estimation arrow_upward

  • Sampling Distribution is a distribution of all the possible values of a statistics for a given size sample selected from a population.
    • It solves the problem of how good is an estimate in a sampling distribution.
  • Sample statistics are used to estimate population parameters.
    • The sample mean estimates the population mean.
    • The sample standard deviation  estimates the population standard deviation.

    Sampling Error arrow_upward

  • The discrepancy between a sample statistics and its population parameter is called Sampling Error.
  • Defining and measuring sampling error is a large part of inferential statistics.

  • Estimation and Confidence Interval arrow_upward

  • A Point Estimate is a single value (statistic) used to estimate a population value (parameter).
    • A point estimator cannot be expected to provide the exact value of the population parameter.
  • An Interval Estimate states the range within which a population parameter probably lies.
    • An interval estimate can be computed by adding and subtracting a margin of error to the point estimate.
  • A Confidence Interval is a range of values within which the population parameter is expected to occur.
    • The two confidence intervals that are used extensively are 95% and 99%.
  • The general form of an interval estimation of a population mean is

  • Where,  = Point Estimate.

  • Interval Estimate of Population Mean arrow_upward

  • When variance  is known

  • Where, 
  •  = Sample Mean.

     = Confidence Coefficient.

     =  value providing an area of   in the upper tail of the standard.

     is the population standard deviation.

     = Sample Size.

  • When variance  is unknown,
  • Where,

     Confidence coefficient

     The  value providing an area of  in the upper tail of a   distribution with  degrees of freedom

    S = Sample Standard Deviation

    Formula for the Sampling Distribution arrow_upward

  • Sample Mean
  • Sample Standard Deviation
  • Standard Error of the Mean is given by

  • Z Score arrow_upward

  • This  score tells us the percentage or proportion of sample sizes  with a sample mean .
  • Normal Population Distribution:
  • Z-value for the sampling distribution of
  • .
  • Where,
  •  Sample Mean.

     Population Mean.

     Population Standard Deviation.     

     Sample Size.

    Z Value for Proportions arrow_upward

  • Where,
  •  is the hypothesized value of population proportion in the null hypothesis,

       p = sample proportion,

       n = sample size,

     = standard deviation of the sampling distribution.

    Null Hypothesis (H0 ) arrow_upward

  • According to the Null Hypothesis (H0 ), the population mean after treatment is same as it was before treatment.
  • Statistical tests mathematically attempt to reject the Null Hypothesis.
  • The equality part of the hypotheses always appears in the Null Hypothesis.

  • Alternative Hypothesis (Ha ) arrow_upward

  • The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment.
  • In a mathematical formulation of the alternative hypothesis there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by Ha .
  • If the null hypothesis is rejected, then we accept the alternative hypothesis.

  • Hypothesis Testing arrow_upward

  • Hypothesis Testing is a technique to determine whether a specific treatment has an effect on the individuals in a population or not.
  • Goal of a hypothesis test is to rule out change (sampling error) as a possible explanation for the results from a research study. 
  • The hypothesis test is used to evaluate the results from a research study in which:
    • A sample is selected from the population.
    • The treatment is administered to the sample.
    • After treatment, the individuals in the sample are measured.
    • If the individuals in the sample are noticeably different from the individuals in the original population, we have evidence that the treatment has an effect.

    Summary of Hypothesis Testing arrow_upward

  • A hypothesis test about the value of a       population mean  must take one of the following three forms (where  is the hypothesized value of the population mean).
  • One-Tailed (lower-tail):


    One-Tailed (upper-tail):




    Purpose of Hypothesis Testing arrow_upward

  • The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief about a parameter.
  • Example:
  • Is a new drug effective in curing a certain disease? A sample of patients is randomly selected.  Half of them are given the drug while the other half are given a placebo. The improvement in the patients’ conditions is then measured and compared.

  • Steps of Hypothesis Testing arrow_upward

    Step 1:
  • Develop the null and alternative hypotheses.
  • Step 2:
  • Specify  and .
  • Step 3:
  • Compute critical Z and actual Z values.
  • Step 4:
  • Use either of the following approaches to make conclusion:
    • p-Value Approach, or
    • Critical Approach

    p-Value Approach arrow_upward

  • The probability value (p-value) of a statistical hypothesis test is the probability of getting a value of the test statistic as extreme as or more extreme than that observed by chance alone, if the null hypothesis H0 , is true.
  • In order to accept or reject the null hypothesis the p-value is computed using the test statistic - Actual Z value.
    • Reject  if the p-value < a.
    • Do not reject (accept)  if the p-value > a.

    Critical Value Approach arrow_upward

  • Use the Z table to find the critical Z value.
  • And, use the equation to find the actual Z statistics.
  • If the actual Z (Z statistics) is in the rejection region, then reject the null hypothesis.
  • Lower tail:
  •  Reject  if actual

    Upper tail: 
  • Reject  if actual

    Critical Region arrow_upward

  • Critical region consists of outcomes that are very unlikely to occur if the null hypothesis is true.
  • That is, the critical region is defined by sample means that are almost impossible to obtain if the treatment has no effect.

  • Test Statistic arrow_upward

  • Test Statistic (z-score) forms a ratio comparing the obtained difference between the sample mean and the hypothesized population mean versus the amount of difference we would expect without any treatment effect.
  • A large value for the test statistic shows that the obtained mean difference is more than would be expected if there is no treatment effect.
    • If it is large enough to be in the critical region, we conclude that the difference is significant or that the treatment has a significant effect.
    • In this case we “reject the null hypothesis”.
    • If the mean difference is relatively small, then the test statistic will have a low value.
    • In this case, we conclude that the evidence from the sample is not sufficient, and the decision is “fail to reject the null hypothesis”.

    Type I and Type II Errors arrow_upward

    Population Condition




    Accept  (Conclude

    Correct Decision

    Type II Error

    Reject  (Conclude

    Type I Error

    Correct Decision

    Thank You from Kimavi arrow_upward

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