Hypothesis Testing Tests on a single sample: If we have the experiment X1, X2,………, Xn representing a random sample from a distribution with Then the hypotheses are:

Hypothesis Testing Standardization of If follow an N(x;0,1)

Hypothesis Testing Then

Hypothesis Testing From diagram we reject H0 if mean less than a or mean greater than b where:

Hypothesis Testing Example 1: A random sample of 100 recorded death in USA the past year, showed an average life span of 71.8 years. Assuming a population standard deviation of 8.9 years, does this seem to indicate that the mean of life span today is greater than 70 years? Use 0.05 level of significance.

Hypothesis Testing Solution: Hypothesis:

Hypothesis Testing Solution:

Hypothesis Testing Solution: 2. Critical region Calculate z as: From table z = 1.645 Calculate z as:

Hypothesis Testing Solution: Then we reject H0 so we conclude that the life span today is greater than 70 years. P-value While z is at rejection area then P-value = (1-P( z < 2.02)) From table P-value = 1- 0.9783 = 0.0217 < alpha so that we reject null hypothesis.

Hypothesis Testing Tests on single sample [variance unknown] When then we should use student t-distribution under the following assumptions: 1.The random variables (X1, X2, ………., Xn) represent random sample from a normal Distribution with

Hypothesis Testing Then the random variable has a t.distribution with (n-1) degree of freedom.

Hypothesis Testing 2. The structure of the test is identical that for sigma is known with the exception that the value sigma in the test is replaced by computed estimate S. 3.The standard normal distribution is replaced by t.distribution as:

Hypothesis Testing Example 2: The Edison Electric Institute has published figures on the number of kilowatt hours used annually by various houses. It is claimed that a vacuum cleaner uses an average of 46 kilowatt hours per year. If a random sample of 12 homes they found vacuum cleaner 42 kilowatt hours per year with standard deviation 11.9 kw/hr. Does this suggest at the 0.05 level of significant that vacuum cleaner use less than 46 kw/hr?

Hypothesis Testing Solution: Hypothesis Critical region: 1- .05 = 0.95

Hypothesis Testing From table then T < -1.796 4. Computations: mean = 42, S =11.9, n=12.

Hypothesis Testing While calculated t is less than tabulated t 1.164 < 1.796 Then We cannot reject H0 and we conclude that the average number of kilowatt hours used is not less than 46.

Hypothesis Testing Example 3: Test the hypothesis that the average content of containers of a particular lubricant is 10 litters if the contents of random sample of 10 containers are: 10.2, 9.7, 10.1, 10.3, 10.1, 9.8,9.9,10.4,10.3,and 9.8 litters use 0.01 level of significant and assume that the distribution of content is normal.

Hypothesis Testing Solution: Hypothesis

Hypothesis Testing 3.Then the critical region is t < -3.25 and t > 3.25 , these t are from table 4. Calculated t:

Hypothesis Testing Solution: Calculated t is less than tabulated t then we don’t reject H0 so that the mean is 10