Hypothesis Testing:�One-Sample Inference

Hypothesis Testing

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Statistical Hypotheses

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Definition

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Statistical hypothesis testing and confidence interval estimation of parameters are the fundamental methods used at the data analysis stage of a comparative experiment, in which the engineer is interested, for example, in comparing the mean of a population to a specified value.

Hypothesis Testing

• Null hypothesis (H0)
• Alternative hypothesis (H1 or Ha)

Null hypothesis (H0)�

• For superiority studies we think for example
• Average systolic blood pressure (SBP) on Drug A is different than average SBP on Drug B
• Null of that? Usually that there is no effect
• Mean = 0
• Sometimes compare to a fixed value so Null
• Mean = 120

How to make conclusions based on data?

• The purpose of most experiments is to prove or disprove a hypothesis.
• This is done by collecting data, analyzing it and drawing a conclusion.
• The original hypothesis is tested against the data to find out whether or not it is right.

Example of a hypothesis

• 636 children from Peru had their lung capacity examined. Response: FEV (Forced Expiratory Volume).
• Scientific question:
• Do boys and girls have different lung capacity?
• Hypothesis:
• H0 : There is no difference in lung capacity for boys and girls.
• We observe:
• Girls : mean(FEV) = 1.54 Boys : mean(FEV) = 1.66.
• Difference = 0.12. How to deal with statistical uncertainty?

Never “Accept” Anything

• Reject the null hypothesis
• Fail to reject the null hypothesis
• Failing to reject the null hypothesis does
• NOT mean the null (H0) is true
• Failing to reject the null means
• Not enough evidence in your sample to reject the null hypothesis

Hypothesis Testing

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Statistical Hypotheses

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null hypothesis

alternative hypothesis

One-sided Alternative Hypotheses

Two-sided Alternative Hypothesis

Hypothesis Testing

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Statistical Hypotheses

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Test of a Hypothesis

• A procedure leading to a decision about a particular hypothesis

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• Hypothesis-testing procedures rely on using the information in a random sample from the population of interest.

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• If this information is consistent with the hypothesis, then we will conclude that the hypothesis is true; if this information is inconsistent with the hypothesis, we will conclude that the hypothesis is false.

Hypothesis Testing

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Tests of Statistical Hypotheses

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Decision criteria for testing H₀:μ = 50 centimeters per second versus H₁:μ ≠ 50 centimeters per second.

Hypothesis Testing

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Tests of Statistical Hypotheses

Definitions

Hypothesis Testing

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Tests of Statistical Hypotheses

Sometimes the type I error probability is called the significance level, or the α-error, or the size of the test.

Hypothesis Testing

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Tests of Statistical Hypotheses

Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

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Hypothesis Testing

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Definition

• The power is computed as 1 - β, and power can be interpreted as the probability of correctly rejecting a false null hypothesis. We often compare statistical tests by comparing their power properties.

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• For example, consider the propellant burning rate problem when

we are testing H ₀ : μ = 50 centimeters per second against H ₁ : μ not equal 50 centimeters per second . Suppose that the true value of the mean is μ = 52. When n = 10, we found that β = 0.2643, so the power of this test is 1 - β = 1 - 0.2643 = 0.7357 when μ = 52.

Hypothesis Testing

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One-Sided and Two-Sided Hypotheses

Two-Sided Test:

One-Sided Tests:

Hypothesis Testing

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P-Values in Hypothesis Tests

Definition

Hypothesis Testing

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P-Values in Hypothesis Tests

Tests on the Mean of a Normal Distribution, Variance Known

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Hypothesis Tests on the Mean

We wish to test:

The test statistic is:

Tests on the Mean of a Normal Distribution, Variance Known

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Hypothesis Tests on the Mean

Reject H₀ if the observed value of the test statistic z₀ is either:

z₀ > z_α/2 or z₀ < -z_α/2

Fail to reject H₀ if

-z_α/2 < z₀ < z_α/2

Tests on the Mean of a Normal Distribution, Variance Known

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Tests on the Mean of a Normal Distribution, Variance Known

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Hypothesis Tests on the Mean

Tests on the Mean of a Normal Distribution, Variance Known

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Hypothesis Tests on the Mean (Continued)

Tests on the Mean of a Normal Distribution, Variance Known

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Hypothesis Tests on the Mean (Continued)

Tests on the Mean of a Normal Distribution, Variance Known

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P-Values in Hypothesis Tests

Tests on the Mean of a Normal Distribution, Variance Known

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Type II Error and Choice of Sample Size

Finding the Probability of Type II Error β (Eq. 9-19)

Tests on the Mean of a Normal Distribution, Variance Known

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Type II Error and Choice of Sample Size

Finding the Probability of Type II Error β

Tests on the Mean of a Normal Distribution, Variance Unknown

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Hypothesis Tests on the Mean

One-Sample t-Test

Tests on the Mean of a Normal Distribution, Variance Unknown

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Hypothesis Tests on the Mean

The reference distribution for H₀: μ = μ₀ with critical region for (a) H₁: μ ≠ μ₀ , (b) H₁: μ > μ₀, and (c) H₁: μ < μ₀.

Tests on the Mean of a Normal Distribution, Variance Unknown

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P-value for a t-Test

The P-value for a t-test is just the smallest level of significance at which the null hypothesis would be rejected.

Notice that t₀ = 2.72 in Example 9-6, and that this is between two tabulated values, 2.624 and 2.977. Therefore, the P-value must be between 0.01 and 0.005. These are effectively the upper and lower bounds on the P-value.

Tests on the Mean of a Normal Distribution, Variance Unknown

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Type II Error and Choice of Sample Size

The type II error of the two-sided alternative (for example) would be

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Hypothesis Test on the Variance

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Hypothesis Test on the Variance

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Hypothesis Test on the Variance

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Hypothesis Test on the Variance

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Type II Error and Choice of Sample Size

For the two-sided alternative hypothesis:

Operating characteristic curves are provided in Charts VIi and VIj:

Hypothesis Tests on the Variance and Standard Deviation of a Normal Distribution

Tests on a Population Proportion

Large-Sample Tests on a Proportion

Many engineering decision problems include hypothesis testing about p.

An appropriate test statistic is

Tests on a Population Proportion

Tests on a Population Proportion

Tests on a Population Proportion

Another form of the test statistic Z₀ is

or

Tests on a Population Proportion

Type II Error and Choice of Sample Size

For a two-sided alternative (Eq. 9-42)

If the alternative is p < p₀ (Eq. 9-43)

If the alternative is p > p₀ (Eq. 9-44)

Tests on a Population Proportion

Type II Error and Choice of Sample Size

For a two-sided alternative (Eq. 9-45)

For a one-sided alternative (Eq. 9-46)

Tests on a Population Proportion

Tests on a Population Proportion

• An engineer who is studying the tensile strength of a steel alloy intended for use in hip replacement knows that tensile strength is approximately normally distributed with σ = 60 psi. A random sample of 12 specimens has a mean tensile strength of x = 3450 psi.

(a) Test the hypothesis that mean strength is 3500 psi. Use α = 0.01.

(b) What is the smallest level of significance at which you would be willing to reject the null hypothesis?

(c) What is the β-error for the test in part (a) if the true mean is 3470?

(d) Suppose that you wanted to reject the null hypothesis with probability at least 0.8 if mean strength μ = 3470. What sample size should be used?

(e) Explain how you could answer the question in part (a) with a two-sided confidence interval on mean tensile strength.

• Medical researchers have developed a new artificial heart constructed primarily of titanium and plastic. The heart will last and operate almost indefinitely once it is implanted in the patient’s body, but the battery pack needs to be recharged about every four hours. A random sample of 50 battery packs is selected and subjected to a life test. The average life of these batteries is 4.05 hours. Assume that battery life is normally distributed with standard deviation σ = 0.2 hour.

• (a) Is there evidence to support the claim that mean battery life exceeds 4 hours? Use α = 0.05.
• (b) What is the P-value for the test in part (a)?
• (c) Compute the power of the test if the true mean battery life is 4.5 hours.
• (d) What sample size would be required to detect a true mean battery life of 4.5 hours if you wanted the power of the test to be at least 0.9?
• (e) Explain how the question in part (a) could be answered by constructing a one-sided confidence bound on the mean life.