Hypothesis and t-tests

HYPOTHESIS ,T Statistic In minitab

-By Manvendra

Not everything you are told is absolutely certain A drug company claims that their miracle drug clears

snoring of 90% of people within 2 weeks, to check this claim a doctor randomly puts 15 patients to test and the results were

i. Why did this happen ? ii. Is the company’s claim false ?

CURED ? YES NO

FREQUENCY 11 4

Hypothesis is a procedure to check possible relationship between 2 or more variables. Make a judgment about difference between sample statistic and hypothesized population parameter.

Population and samples

PREVIOUS EXAMPLE :

: MIRACLE DRUG CLEARS SNORING OF 90% OF PATIENT : THE ABOVE IS NOT TRUE

Types of hypothesis :

Null/Nil hypothesis (H0)• Statistical hypothesis

Alternative hypothesis(H1 or Ha)

• Empirical hypothesis

TYPES OF ERROR : Type I error(α)= When the researcher rejects a null hypothesis

when it is true. The probability of committing a Type I error is called the significance level.

Type II error(β)=When the researcher fails to reject a null hypothesis that is false. The probability of no t c om m i t t i ng a Type II error is called the Power of the test.

2-TAIL TEST

Significance level indicate percentage of sample mean that is outside certain limits.

Example, In the United States watch an average of 3 hours of TV per week. To test whether this claim is true, we record the time (in hours) that a group of 20 American children (the sample), among all children in the United States (the population), watch TV. The mean we measure for these 20 children is a sample mean

p-value: The strength of evidence in support of a null hypothesis is measured by the P-value .The smaller the p-value the more significant is the data. p-values can only be used to reject the hypothesis and not to consider them. Lies between 0 and 1.

Example,A coin is tossed 5 times to check if the coin is unbiased.

Solution:H0 :The coin is fair v/s H1 :The coin is unfair The probability of 2 outcomes can be H or T .

The test statistic is 5 by Bernoulli trial, the p-value is

Here p-value<α , Reject H0.

P is low, so the null must go.

For reporting results (e.g. for Minitab),i. Compute from the observations the observed

value T obs. of the test statistic T (generally at α=5%)

ii. Calculate the p-valueiii. Decision criterion . If p value<α the decision is to

Reject H0.

STEPS FOR HYPOTHESIS :

T-statisticProperties and Applications

Interpretation in Minitab

T Test :

Introduced by William Sealy Gosset The basic need of a T-test is to check the null

hypothesis if the means of 2 sample groups are equal. It is limited to 2 groups

n<30 and if not it’s assumed to be distributed normally The standard deviation of population is unknown

Student’s T-test

• To see if a sample mean is significantly different from a population mean. Only the sample s.d. is known.

• Also used for large samples1 sample

test

• To test if 2 samples representing different populations have same mean

• Independent of each other2 sample

test• 1 sample from which 2 measurements

are made i.e. Dependent• Used to compare the difference,

BEFORE & AFTER of same sample

Paired test

TYPES OF T-TEST :

1 sample test 2 sample test Paired test

1)A curious student wants to check if the human body temp. is actually 98.6 F At 5% L.O.S

2) The amount of coffee (in ounces) filled by amachine in six randomly picked jars: 15.7, 15.9, 16.3, 16.2, 15.7 and 15.9. Is the true mean amount of coffee in a jar is 16 ounces?

1) To study the effect of drug with diet alone and diet and drug considering from 2 different population

2) Below are given the gain in weights (in lbs) of pigs fed on two diets A and BDiet A: 25, 32, 30, 43, 24, 14, 32, 24, 31, 31, 35, 25Diet B: 44, 34, 22, 10, 47, 31, 40, 30, 32, 35, 18, 21, 35, 29Test, if the two diets differ significantly as regards their effect on increase in weight

1) Mean time taken to sleep decreases when reading Hawthorne before sleep. Data has been obtained for Hawthorne and without Hawthorne of same population within a week’s duration

2) Eleven school boys were given a test in mathematics. Do the marks give evidence that the student’s have benefited by the extra coaching? Marks in test-1: 23, 20, 19, 21, 18, 20, 18, 17, 23, 16, 19Marks in test-2: 24, 19, 22, 18, 20, 22, 20, 20, 23, 20

EXAMPLES :

USING MINITAB

1-sample t-test

2-sample t-test

Paired Sample t-test

Paired Sample t-test