Hypothesis TestingHypothesis is making an assumption. In Statistics a Hypothesis or an assumption is taken first and then the Hypothesis is tested as how accurate it is or not. Hypothesis testing is a study based on statistical accuracy of an experiment. If the result is positive i.e. if the assumption is correct or approximate, then it is called Statistically Significant.

There are two types of statistical hypotheses

Null Hypothesis: Denoted by H0, it is actually an assumption that the sample observations are purely from chance.

Alternate Hypothesis: Denoted by H1 or Ha, it assumes that the sample is influenced by a non-random cause.

When we test a hypothesis we proceed as follows:

1. Formulate the null and alternative hypothesis.

2. Level of significance has to be determined.

3. The size of the sample has to be chosen.

4. With the help of the table, determine if the z score falls within the acceptance region.

5. Decide to

a. Reject the null hypothesis and accept the alternative hypothesis or

b. Fail to reject the null hypothesis and state that there is not enough evidence to support the alternative hypothesis.

Statisticians follow a formal process to determine whether to accept or reject null-hypothesis, based on sample data. This process is called hypothesis testing and consists of four steps.

State the hypotheses: The first step involves stating the null and alternate hypotheses. The hypotheses have to be stated in such a way that they are mutually exclusive.

Formulate an analysis plan: The analysis plan describes how to use sample data to evaluate the null hypothesis. This evaluation focuses around a single test statistic.

Analyze sample data: Find the value of the test statistic (mean score, proportion, t-score, z-score, etc.) described in the analysis plan

Interpret results: Apply the decision rule described in the analysis plan. If the value of the test statistic is unlikely, based on the null-hypothesis, reject the null hypothesis.

Decision Errors. Two types of errors can result from a hypothesis test

Type I Error: A Type I Error occurs when the researcher rejects a null-hypothesis when it is actually true. The probability of committing a Type I error is called the significance level. This probability is called alpha and is often denoted by α.

Type II Error: A Type II Error occurs when the researcher fails to reject a null-hypothesis when it is false. The probability of committing a Type II Error is called Beta and is often denoted by β.

Decision Rules

The analysis plan includes decision rules for rejecting the null-hypothesis. In practice, statisticians describe these decision rules in two-ways with reference to a P-value or with reference to a region of acceptance.

P-value: The strength of evidence in support of a null-hypothesis is measured by the p-value. Suppose the test statistic is equal to S. The P-value is the probability of observing a test statistic as extreme as S, assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis.

Region of acceptance: The region of acceptance is a range of values. If the test statistic falls under the region of acceptance the null-hypothesis is not rejected. The region of acceptance is defined so that the chance of making a Type I error is equal to the significance level.

The set of values outside the region of acceptance is called the region of rejection. If the test statistic falls under the region of rejection, the null-hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance.

Example Problems

1. A newspaper article stated that students at a particular university spend an average of $95 on soft drinks. A student investigator who believed this average was too high polled a random sample of 50 students and found that x̄ = $92.25 and s = $10. Use these results to test at a 5% significance level the statement made by the newspaper.

Given: = $95 ?, n = 50, x̄ = $92.25, s = $10.00, 5% significance

H0 : = $95 H1 : < $95 (Shown to be 'too high' only at lower end)

Significance level: 5%

Critical value: Large sample, unknown, z-table, 5%, 1 tail, 1.64

Test statistic: z=

|̄x−μ|s /√n

⇒|92. 25−95 .00|10/√50

= 2 .751. 414

=1 .94

Conclusion: Test statistic larger than critical value so null hypothesis rejected. The mean amount spent is less than $95.

2. A coin is suspected of being biased. It is tossed 200 times and 114 heads occur. Carry out a hypothesis test to see if the coin is indeed biased at 1% significance.

Given: = 50%?, p=114

200×100=57 %

, n = 200, 1% significance

H0 : = 50% H1 : 50% (Shown to be biased at either extreme)

Significance level: 1%

Critical value: Large sample, proportion, z-table, 1%, 2 tail, 2.58

Test statistic:

z=|p−π|

√ π (100−π )n

⇒|57−50|

√50×50200

= 73 . 5

=1. 98

Conclusion: Test statistic less than critical value so null hypothesis not rejected. The coin could be fair.

3. Eleven cartons of sugar, each nominally containing 1 Kgs, were randomly selected from a large batch of cartons. The weights of sugar they contained were:

1.02 1.05 1.08 1.03 1.00 1.06 1.08 1.01 1.04 1.07 1.00 Kgs

Does this support the hypothesis, at 5%, that the mean weight for the whole batch is over 1.00 kg?

Given: = 1.00 kg? From the sample: n = 11, x̄ = 1.04 Kgs, s = 0.0303 lb, 5% significance

H0 : = 1.00 kg H1 : > 1.00 kg (Shown to be 'over' only at higher end)

Significance level: 5%

Critical value: Small sample, unknown, t-table, 5%, 1 tail, 10 deg. of free., 1.81

Test statistic: t=

|̄x−μ|s /√n

⇒|1 .04−1. 00|0 . 0303/√11

= 0. 040. 00914

=4 . 38

Conclusion: Test statistic larger than critical value so null hypothesis rejected. The mean weight could be over 1.00 kg.