Hypothesis testing

Сентябрь 13, 2022

Содержание

2. Steps in Hypothesis Testing
3. 1st step: Stating the hypotheses
5. 2nd step: Identifying the appropriate test statistic and its probability distribution
6. 3rd: Specifying the significance level The level of significance reflects how much sample evidence we require
7. 4th: Stating the decision rule A decision rule consists of determining the rejection points (critical values)
8. 5th: Collecting the data and calculating the test statistic Collecting the data by sampling the population
9. p-Value The p-value is the smallest level of significance at which the null hypothesis can be
10. Tests Concerning a Single Mean For hypothesis tests concerning the population mean of a normally distributed
12. The z-Test Alternative
14. Tests Concerning Differences between Means When we want to test whether the observed difference between two
17. Tests Concerning Mean Differences In tests concerning two means based on two samples that are not
20. Tests Concerning a Single Variance In tests concerning the variance of a single, normally distributed population,
21. Rejection Points for Hypothesis Tests on the Population Variance.
22. Tests Concerning the Equality (Inequality) of Two Variances For tests concerning differences between the variances of
25. NONPARAMETRIC INFERENCE A parametric test is a hypothesis test concerning a parameter or a hypothesis test
26. Tests Concerning Correlation: The Spearman Rank Correlation Coefficient The Spearman rank correlation coefficient is essentially equivalent
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Steps in Hypothesis Testing

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1st step: Stating the hypotheses

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2nd step: Identifying the appropriate test statistic and its probability distribution

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3rd: Specifying the significance level
The level of significance reflects how much

sample evidence we require to reject the null. Analogous to its counterpart in a court of law, the required standard of proof can change according to the nature of the hypotheses and the seriousness of the consequences of making a mistake. There are four possible outcomes when we test a null hypothesis:
The probability of a Type I error in testing a hypothesis is denoted by the Greek letter alpha, α. This probability is also known as the level of significance of the test.

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4th: Stating the decision rule
A decision rule consists of determining the

rejection points (critical values) with which to compare the test statistic to decide whether to reject or not to reject the null hypothesis. When we reject the null hypothesis, the result is said to be statistically significant.

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5th: Collecting the data and calculating the test statistic
Collecting the data

by sampling the population
To reject or not
The first six steps are the statistical steps. The final decision concerns our use of the statistical decision.
The economic or investment decision takes into consideration not only the statistical decision but also all pertinent economic issues.

6th: Making the statistical decision

7th: Making the economic or investment decision

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p-Value
The p-value is the smallest level of significance at which the

null hypothesis can be rejected. The smaller the p-value, the stronger the evidence against the null hypothesis and in favor of the alternative hypothesis. The p-value approach to hypothesis testing does not involve setting a significance level; rather it involves computing a p-value for the test statistic and allowing the consumer of the research to interpret its significance.

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Tests Concerning a Single Mean
For hypothesis tests concerning the population mean

of a normally distributed population with unknown (known) variance, the theoretically correct test statistic is the t-statistic (z-statistic). In the unknown variance case, given large samples (generally, samples of 30 or more observations), the z-statistic may be used in place of the t-statistic because of the force of the central limit theorem.
The t-distribution is a symmetrical distribution defined by a single parameter: degrees of freedom. Compared to the standard normal distribution, the t-distribution has fatter tails.

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The z-Test Alternative

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Tests Concerning Differences between Means
When we want to test whether the

observed difference between two means is statistically significant, we must first decide whether the samples are independent or dependent (related). If the samples are independent, we conduct tests concerning differences between means. If the samples are dependent, we conduct tests of mean differences (paired comparisons tests).
When we conduct a test of the difference between two population means from normally distributed populations with unknown variances, if we can assume the variances are equal, we use a t-test based on pooling the observations of the two samples to estimate the common (but unknown) variance. This test is based on an assumption of independent samples.

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Tests Concerning Mean Differences
In tests concerning two means based on two

samples that are not independent, we often can arrange the data in paired observations and conduct a test of mean differences (a paired comparisons test). When the samples are from normally distributed populations with unknown variances, the appropriate test statistic is a t-statistic. The denominator of the t-statistic, the standard error of the mean differences, takes account of correlation between the samples.

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Tests Concerning a Single Variance
In tests concerning the variance of a

single, normally distributed population, the test statistic is chi-square (χ2) with n − 1 degrees of freedom, where n is sample size.

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Rejection Points for Hypothesis Tests on the Population Variance.

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Tests Concerning the Equality (Inequality) of Two Variances
For tests concerning differences

between the variances of two normally distributed populations based on two random, independent samples, the appropriate test statistic is based on an F-test (the ratio of the sample variances).

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NONPARAMETRIC INFERENCE
A parametric test is a hypothesis test concerning a parameter

or a hypothesis test based on specific distributional assumptions. In contrast, a nonparametric test either is not concerned with a parameter or makes minimal assumptions about the population from which the sample comes.
A nonparametric test is primarily used in three situations: when data do not meet distributional assumptions, when data are given in ranks, or when the hypothesis we are addressing does not concern a parameter.

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Tests Concerning Correlation: The Spearman Rank Correlation Coefficient
The Spearman rank correlation coefficient

is essentially equivalent to the usual correlation coefficient calculated on the ranks of the two variables (say X and Y) within their respective samples. Thus it is a number between −1 and +1, where −1 (+1) denotes a perfect inverse (positive) straight-line relationship between the variables and 0 represents the absence of any straight-line relationship (no correlation). The calculation of rS requires the following steps:

Hypothesis testing

Содержание

Steps in Hypothesis Testing

1st step: Stating the hypotheses

2nd step: Identifying the appropriate test statistic and its probability distribution

3rd: Specifying the significance levelThe level of significance reflects how much

4th: Stating the decision ruleA decision rule consists of determining the

5th: Collecting the data and calculating the test statisticCollecting the data

p-ValueThe p-value is the smallest level of significance at which the

Tests Concerning a Single MeanFor hypothesis tests concerning the population mean

The z-Test Alternative

Tests Concerning Differences between MeansWhen we want to test whether the

Tests Concerning Mean DifferencesIn tests concerning two means based on two

Tests Concerning a Single VarianceIn tests concerning the variance of a