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 45b.More than 50

Example 2

The achievement sores for a college entrance examination are normally distributed with the mean 75 and standard deviation equal to 10. What fraction of the scores would one expect to lie between 70 and 90.

Sampling Distribution

• Distribution of all possible sample statistics

Population All Possible Samples

Sample Means

1, 2, 3, 4

1, 2, 3 2.001, 2, 4 2.333, 4, 1 2.672, 3, 4 3.00

μ = 2.5;σ = 1.18 n = 3 μxbar = 2.5

Central Limit Theorem

Given a distribution with a mean μ and variance σ², the sampling distribution

of the mean approaches a normal distribution 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 and standard 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 TESTINGNormal 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

Two-tailed test

• Alternative hypothesis does not specify a directional difference for the parameter of interest

– H0: μ = 10 v. Ha: μ ≠ 10

– 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

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.

Testing a Hypothesis on thePopulation Mean

H0 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 driven on 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.