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Normal Distribution and

Hypothesis Testing

STR1K



Characteristics

• Bell-shaped, depends on standard deviation

• Continuous distribution

•

Unimodal• Symmetric about the vertical axis through the

mean μ

•

Approaches the horizontal axis asymptotically• Total area under the curve and above the

horizontal is 1



Characteristics

• Approximately 68% of observations fall within

1σ from the mean

• Approximately 95% of observations fall within

2σ from the mean

• Approximately 99.7% of observations fall

within 3σ from the mean



68.27%

95.45 %

99.73%



Standard Normal Distribution

• Special type of normal distribution where μ =0

• Used to avoid integral calculus to find the area

under the curve

• Standardizes raw data

• Dimensionless Z-score

Z = X - μ0

σ



Example 1

Given the normal distribution with

μ = 49 and σ = 8, find the probability

that X assumes a value:

a. Less than 45

b. More than 50



Example 2

The achievement sores for a college

entrance examination are normally

distributed with the mean 75 andstandard deviation equal to 10. What

fraction of the scores would one

expect to lie between 70 and 90.



Central Limit Theorem

Given a distribution with a mean μ and

variance σ², the sampling distribution

of the mean approaches a normaldistribution with a mean (μ) and a

variance σ²/N as N, the sample

size, increases.



Characteristics

• The mean of the population and the mean of

the sampling distribution of means will always

have the same value.

• The sampling distribution of the mean will be

normal regardless of the shape of the

population distribution.



N(70, 16)



N(70,1)



N(70,.25)



Characteristics

• As the sample size increases, the distribution

of the sample average becomes less and less

variable.

• Hence the sample average Xbar approaches the

value of the population mean μ.



Example 3

An electrical firm manufactures light

bulbs that have a length of life

normally distributed with mean andstandard deviation equal to 500 and

50 hours respectively. Find the

probability that a random sample of

15 bulbs will have an average life

ofless than 475 hours.



HYPOTHESIS TESTING

Normal Distribution and Hypothesis Testing



Hypothesis Testing

• A hypothesis is a conjecture or assertion

about a parameter

• Null v. Alternative hypothesis

– Proof by contradiction

– Null hypothesis is the hypothesis being tested

– Alternative hypothesis is the operational

statement of the experiment that is believed to be

true



One-tailed test

• Alternative hypothesis specifies a one-directional

difference for parameter

– H0: μ = 10 v. Ha: μ < 10

– H0: μ = 10 v. Ha: μ > 10

– H0: μ1 - μ2 = 0 v. Ha: μ1 - μ2 > 0

– H0: μ1 - μ2 = 0 v. Ha: μ1 - μ2 < 0



Critical Region

• Also known as the “rejection region”

• Critical region contains values of the test

statistic for which the null hypothesis will be

rejected

• Acceptance and rejection regions are

separated by the critical value, Z.



Type I error

• Error made by rejecting the null hypothesis

when it is true.

• False positive

• Denoted by the level of significance, α

• Level of significance suggests the highest

probability of committing a type I error



Type II error

• Error made by not rejecting (accepting) the

null hypothesis when it is false.

• False negative

• Probability denoted by β



Decision H0 true H0 false

Reject H0

Type I error

(α)

Correct

decision

(1-β)

Accept H0

Correct

decision(1-α)

Type II error

(β)



Notes on errors

• Type I (α) and type II errors (β) are related. A

decrease in the probability of one, increases

the probability in the other.

• As α increases, the size of the critical region

also increases

• Consequently, if H0 is rejected at a low α, H0

will also be rejected at a higher α.



Make a decision. Reject H0 if the value of the test statistic belongs tothe critical region.

Collect the data and compute the value of the test statistic from thesample data

Select the appropriate test statistic and establish the critical region

Choose the level of significance, α

State the null hypothesis (H0) and the alternative hypothesis (Ha)



Testing a Hypothesis on the

Population MeanH0 Test Statistic Ha Critical Region

σ known

μ = μ0

μ < μ0

μ > μ0

μ ≠ μ0

z < -zα

z > zα

|z| > zα/2

σ unknown

μ = μ0

μ < μ0

μ > μ0

μ ≠ μ0

t < -tα

t > tα

|t| > tα/2

Z = X - μ0

σ /√n

t = X - μ0

S /√n

υ = n - 1



critical value

test statistic

Reject H0



critical value

test statistic

Do not reject H0



Example 4

It is claimed that an automobile is drivenon the average of less than 25,000 km per

year.

To test this claim, a random sample of 100

automobile owners are asked to keep a

record of the kilometers they travel.

Would you agree with this claim if the

random sample showed an average of 23,500 km and a standard deviation of

3,900 km? Use 0.01 level of significance.

hypothesis-testing-1228306543459708-9

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