# Chapter 10 : Sampling Distribution and Hypothesis Testing

### Topics covered in this snack-sized chapter:

#### 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 ofPopulation 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):  ##### Two-Tailed:  #### 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 Conclusion True  False Accept (Conclude Correct Decision Type II Error Reject (Conclude Type I Error Correct Decision

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