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# Hypothesis Testing

## by Kent Kawashima, Teacher at Philippine Science High School on Dec 03, 2008

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## Hypothesis TestingPresentation Transcript

• 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 1 σ from the mean
• Approximately 95% of observations fall within 2 σ from the mean
• Approximately 99.7% of observations fall within 3 σ 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 - μ 0 σ
• Example 1
• Given the normal distribution with μ = 49 and σ = 8, find the probability that X assumes a value:
• Less than 45
• 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 2.00 1, 2, 4 2.33 3, 4, 1 2.67 2, 3, 4 3.00 μ = 2.5; σ = 1.18 n = 3 μ xbar = 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 X bar 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 TESTING
• Normal Distribution and Hypothesis Testing
• Hypothesis Testing
• A hypothesis is a conjecture or assertion about a parameter
• Null v. Alternative hypothesis
• 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
• H 0 : μ = 10 v. H a : μ < 10
• H 0 : μ = 10 v. H a : μ > 10
• H 0 : μ 1 - μ 2 = 0 v. H a : μ 1 - μ 2 > 0
• H 0 : μ 1 - μ 2 = 0 v. H a : μ 1 - μ 2 < 0
• Two-tailed test
• Alternative hypothesis does not specify a directional difference for the parameter of interest
• H 0 : μ = 10 v. H a : μ ≠ 10
• H 0 : μ 1 - μ 2 = 0 v. H a : μ 1 - μ 2 ≠ 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 H 0 is rejected at a low α , H 0 will also be rejected at a higher α .
•
• Testing a Hypothesis on the Population Mean Z = X - μ 0 σ /√n t = X - μ 0 S /√n υ = n - 1 H 0 Test Statistic H a Critical Region σ known μ = μ 0 μ < μ 0 μ > μ 0 μ ≠ μ 0 z < -z α z > z α |z| > z α /2 σ unknown μ = μ 0 μ < μ 0 μ > μ 0 μ ≠ μ 0 t < -t α t > t α |t| > t α /2
• critical value test statistic Reject H 0
• critical value test statistic Do not reject H 0
• 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.