Hypothesis Testing

What is hypothesis

“An Assumption or Statement when statistically tested based on the data collected”.

Hypothesis

Null Hypothesis -They are proposed with an intent to receiving a rejection. These are denoted as Ho. Alternate HypothesisCan cover a whole range of value rather than a single point and denoted by H1.

Steps in testing of Hypothesis 1. Formulate the HypothesisA criminal trial is an example of hypothesis testing without the statistics. In a trial a jury must decide between two hypotheses. The null hypothesis is

H0: The defendant is innocent

The alternative hypothesis or research hypothesis isH1: The defendant is guilty

The jury does not know which hypothesis is true. They must make a decision on the basis of evidence presented.

Non-Statistical Hypothesis TestingIn the language of statistics convicting the defendant is called rejecting the null hypothesis in favor of the alternative hypothesis. That is, the jury is saying that there is enough evidence to conclude that the defendant is guilty (i.e., there is enough evidence to support the alternative hypothesis).

If the jury acquits it is stating that there is not enough evidence to support the alternative hypothesis. Notice that the jury is not saying that the defendant is innocent, only that there is not enough evidence to support the alternative hypothesis. That is why we never say that we accept the null hypothesis, although most people in industry will say “We accept the null hypothesis”, hence we say that we fail to reject null hypothesis.

Null Hypothesis

“A statement in which no difference or effect is expected. It is always required to test null hypothesis”.

Null Hypothesis is also represented by certain value of the population parameter (e.g. µ,π,σ). Null Hypothesis may be rejected, but it can never be accepted based on single test.

For eg: A dept. store is considering the introduction of an internet shopping services, only in case if more than 40% of the internet users shop via internet.

Ho: π ≤ 0.04H1: π ˃ 0.04If in case Ho is not rejected , then services will only be introduced after getting additional evidence.This test of hypothesis is a one tail as it is expressed directionally. i.e level of significance is more than 0.04.

On the other hand, suppose the researcher wanted to determine whether the proportion of internet users who shop via the internet is different from 40%, than the two tail test will be required.

Ho: π = 0.04H1: π ≠ 0.04

Differences between One and Two-tail test 1.A one-tailed test looks for an increase or decrease in the parameter whereas a two-tailed test looks for any change in the parameter (which can be any change- increase or decrease).

2.When you conduct a test of statistical significance, whether it is from a correlation, an ANOVA, a regression or some other kind of test, you are given a p-value somewhere in the output. If your test statistic is symmetrically distributed, you can select one of three alternative hypotheses. Two of these correspond to one-tailed tests and one corresponds to a two-tailed test. However, the p-value presented is (almost always) for a two-tailed test.that means statistical software always report two tail results

3.One-tailed tests are used for asymmetric distributions that have a single tail, such as the chi-squared distribution, which are common in measuring goodness-of-fit, or for one side of a distribution that has two tails, such as the normal distribution, which is common in estimating location; this corresponds to specifying a direction. Two-tailed tests are only applicable when there are two tails, such as in the normal distribution, and correspond to considering either direction significant.

Step-2 Select an Appropriate TestTest Statistic“ It Measures of how close the sample has come to the null hypothesis. It often follows a well-known distribution, such as the normal, t, or chi-square”.ɀ

𝜎𝜌= √π﴾1-π﴿/n

Step-3: Choose Level of SignificanceThere are two possible errors.1. Type-I error2. Type- II errorA Type I error occurs when we reject a true null hypothesis. That is, a Type I error occurs when the jury convicts an innocent person.

The probability of a Type I error is denoted as α =0.40(Greek letter alpha)

Α= level of significance(probability of making type-I error)

Type-II Error

A Type II error occurs when we don’t reject a false null hypothesis [accept the null hypothesis]. That occurs when a guilty defendant is acquitted.

The probability of a type II error is β ≤0.20 (Greek letter beta).The two probabilities are inversely related. Decreasing one increases the other, for a fixed sample size.(1-β)= Power of a statistical test.

Non-statistical Hypothesis Testing…A Type I error occurs when we reject a true null hypothesis (i.e. Reject H0 when it is TRUE)

A Type II error occurs when we don’t reject a false null hypothesis (i.e. Do NOT reject H0 when it is FALSE)

Reject

The critical concepts are theses:1. There are two hypotheses, the null and the alternative

hypotheses.2. The procedure begins with the assumption that the null

hypothesis is true.3. The goal is to determine whether there is enough evidence

to infer that the alternative hypothesis is true, or the null is not likely to be true.

4. There are two possible decisions:Conclude that there is enough evidence to support the

alternative hypothesis. Reject the null.Conclude that there is not enough evidence to support

the alternative hypothesis. Fail to reject the null.

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Step-4: Collect Data and Calculate Test StatisticStep-5: Compare the probability and make decisionStep-6: Research Conclusion

Elements of a Hypothesis Test• Test Statistic - Difference between the Sample means, scaled

to number of standard deviations (standard errors) from the null difference of 0 for the Population means:

• Rejection Region - Set of values of the test statistic that are consistent with HA, such that the probability it falls in this region when H0 is true is a (we will always set a=0.05)

P-value (aka Observed Significance Level)• P-value - Measure of the strength of evidence the sample data

provides against the null hypothesis:P(Evidence This strong or stronger against H0 | H0 is true)

Criteria for Accepting/ Rejecting Null Hypothesis

S/No Alternative Hypothesis Reject Null Hypothesis, if

Reject Null Hypothesis, if

1 µ<µo Z < -Zα Z ≥ -Zα

2 µ˃µo Z ˃ Zα Z ≤ Zα

3 µ ‡ µo Z < -Zα/2 or Z ˃ Zα/2

-Zα/2≤ Z≤ Zα/2

Hypothesis Testing Concerning Mean Value• Case of large Sample: through Z-Test

Z= X- µo σ/√n

X = Sample Meanµo =Null hypothesisσ=Population meann=sample size

For e.g

Testing Null HypothesisThe two possible decisions that can be made:

1. Conclude that there is not enough evidence to support the alternative hypothesis (also stated as: failing to reject the null hypothesis in favor of the alternative)• NOTE: we do not say that we accept the null hypothesis if a

statistician is around…

2. Conclude that there is enough evidence to support the alternative hypothesis (also stated as: reject the null hypothesis in favor of the alternative)

Alternate Approach to test the hypothesisHere we find the P valuep = P(Z>Zα) α= 0.05

Refer the above table

For e.g• On a tying test, a random sample of 36 students of a school

averaged 73.6 words with a standard deviation of 8.10 words /min. Test an employers’ claim that the school students average less than 75.0 words /min using the 5 % level of significance.

Ho =µ = 75H1= µ< 75

X = 73.6, S= 8.10, N=36, α= 0.05 ( as sample size is large, and standard deviation is unknown, Z test is appropriate)Z= X- µo σ/√n

Z= 73.6-75/1.35= -1.4/1.35=-1.04 As the critical value has to be more than -1.04 we fail to reject null hypothesis in favor of Alternate hypothesis.

This can be checked by deriving p value alsop = Z < -1.04

p= 0.05 -0.03508

= 0.01492Hence failed to reject null hypothesis

Reasons for Hypothesis Testing

Testing

For Association

ForDifference

Distribution Mean Proportions Medians/ Ranking

Hypothesis Testing

Hypothesis Testing

Parametric Non- Parametric• Variables are measured on at least Variables are measured • an Interval Scales on a normal/ ordinal scale• T, z test is applied Chi Square test is applied

Parametric Test

Parametric Test provides inferences for making statement about the means of parent population. A t test is commonly used

t, test : “ A univariate hypothesis test using the t distribution, which is used when the standard deviation is unknown and the sample size is small”

This test is based on students’ t statistic, with an assumption that variables are normally distributed and mean is known and population variance is estimated from the sample.

To find Association

Chi Square Test