Hypothesis Testing

Hypothesis testing is used for testing a claim about a parameter in a population, using data measured in a sample. Hypothesis Testing is sometimes referred to as significance testing. Hypothesis testing aims to make a statistical conclusion about accepting or rejecting the hypothesis.

The below steps are involved in statistical hypothesis testing

1. Formulate the Null Hypothesis (Ho) and Alternative Hypothesis (Ha) — In general, null hypothesis is the commonly accepted fact and proposes that no statistical significance exists in a given set of observations. The Alternate hypothesis is the statement which contradicts null hypothesis.
2. State the significance level — Level of significance is the probability with which we will reject a null hypothesis when it is true. Level of significance is defined by alpha (a). Confidence Interval (CI) = 1-a
3. Calculate test statistic.
4. Decide to Reject or Retain Null hypothesis — If the test statistic falls within the accepted region retain Retail Null hypothesis, if it falls in the rejection region reject the hypothesis

Type I Error and Type II Error

Type I error is rejecting the Null Hypothesis (Ho ) when in reality Ho is true

Type II error is rejecting the Alternative Hypothesis (Ha) when in reality Ha is true

The type I error level is the significance level denoted by alpha (a). Since Type I error is more serious error we generally set small values of alpha.

Hypothesis Testing — Error Types

Rejection Regions

Critical values, which mark the cutoffs for the rejection region, can be identified for any level of significance. The rejection region is the region beyond a critical value in a hypothesis test. When the value of a test statistic is in the rejection region, we decide to reject the null hypothesis; otherwise, we retain the null hypothesis

Type of Tests

1. Two Tailed

Ho: µ = µ0

Ha: µ ≠ µ0

Critical Regions consists of both tails of sampling distribution of the test statistic.

2. One tailed (left sided)

Ho: µ = µ0

Ha: µ < µ0

Critical Region consists of left tail of the sampling distribution of the test statistic.

3. One tailed (right sided)

Ho: µ = µ0

Ha: µ > µ0

Critical Region consists of right tail of the sampling distribution of the test statistic.

Commonly used critical Values for one and two tailed tests:

For the below 2 commonly used levels of significance (a)

a = 0.05: One Tailed Test +- 1.645, Two Tailed Test+- 1.96

a = 0.01: One Tailed Test +- 2.33, Two Tailed Test +- 2.58

Two Tailed Test

Z Test

Used when the sample size n > 30 (If the sample size is small use t test instead).

Z statistic is calculated as:

Z test

Around 68% of the elements have a z-score between -1 and 1, 95% have a z-score between -2 and +2

Illustrative example.

The problem statement:

Suppose you need to investigate a claim that in a faraway Village Choco land, every 10-year-old kid consume 800 calories of chocolate per day. You select a random sample of 36 10-year-old kids. The average of the daily calories consumed by these 36 kids is 806 calories. (Assume we know that the population standard deviation of calories consumed by the 10-year-old kids is 12 calories)

Solution:

a)State the Null Hypothesis and Alternate Hypothesis

Null Hypothesis: Ho: x = 800

Alternate Hypothesis: Ha: µ ≠ µ0

This will be a 2 tailed test

b)State the significance level

Significance Level (a) is set at 0.05

c)Test Statistic

Sample size n > 30. We will use the z test.

Test Statistic Z = (806– 800)/(12/root 36) = 3

The calculated test statistic is >2

d)Decision to reject or retain the hypothesis

Null hypothesis is rejected. Alternate hypothesis is chosen.

Therefore, every 10-year-old kid do not consume 800 calories of chocolate per day

Hope this article was helpful to you. Thank you.