Chapter 10 : Sampling Distribution and Hypothesis Testing
Sampling Distribution is a distribution of all the possible values of a statistics for a given size sample selected from a population.
Topics covered in this snack-sized chapter:
Sample statistics are used to estimate population parameters.
- It solves the problem of how good is an estimate in a sampling distribution.
- The sample mean estimates the population mean.
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.
A Point Estimate is a single value (statistic) used to estimate a population value (parameter).
- The sample standard deviation estimates the population standard deviation.
An Interval Estimate states the range within which a population parameter probably lies.
- A point estimator cannot be expected to provide the exact value of the population parameter.
A Confidence Interval is a range of values within which the population parameter is expected to occur.
- An interval estimate can be computed by adding and subtracting a margin of error to the point estimate.
The general form of an interval estimation of a population mean is
- The two confidence intervals that are used extensively are 95% and 99%.
Where, = Point Estimate.
When variance is known
= 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,
The value providing an area of in the upper tail of a distribution with degrees of freedom
S = Sample Standard Deviation
Sample Standard Deviation
Standard Error of the Mean is given by
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
Population Standard Deviation.
is the hypothesized value of population proportion in the null hypothesis,
p = sample proportion,
n = sample size,
= standard deviation of the sampling distribution.
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.
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 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.
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).
The purpose of hypothesis testing is to determine whether there is enough statistical evidence in favor of a certain belief about a parameter.
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.
Develop the null and alternative hypotheses.
Specify and .
Compute critical Z and actual Z values.
Use either of the following approaches to make conclusion:
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.
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.
- Do not reject (accept) if the p-value > a.
Reject if actual
Reject if actual
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 (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 II Error
Type I Error