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Stat 305, Fall 2014 Name

Chapter 6 Part II: Hypothesis Testing for

Motivating Example (Example 1)

Suppose you are the manufacturer of construction equipment. You make 14

wire rope and

need to determine how much weight it can hold before breaking so that you can label itclearly and avoid lawsuits. Suppose a SRS of size n= 25 lengths of 1

4 wire rope resulted

in a sample average of 15 tons, and that we somehow know that 2 = 36.One buyer will only buy your wire rope if the true mean breaking strength is above 12 tons.

If the true breaking strength was in fact 12 tons (i.e. = 12), what would thedistribution of the sample mean be?

Given this distribution, what is the probability of obtaining a sample of size n = 25with mean of at least 15 tons?

Based on this probability, what would you tell the buyer?

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Formal Hypothesis Testing

Congratulations! You just completed your first hypothesis test. Now we need the formalstructure of a hypothesis test (also called a significance test). We just need to follow these5 steps.[DISCLAIMER: This is not the same as the 5 steps from the book!!!]

Step 1: State the null and alternative hypotheses. The Null Hypothesis (H0) is a statement of the parameter being equal to some

number (usually a number that is what we want to disprove, either based on historyor experience)

H0: = 0 (where 0 is just a number.)

TheAlternative Hypothesis(HA,orH1) is a statement of opposition to the null hy-pothesis (usually reflects what we really believe is true about the parameter comparedto H0)

HA : < 0 (where 0 is the same number as in H0.)

HA : > 0 HA : =0

Step 2: Compute the test statistic and state its distribution.

A test statistic is a formula that summarizes the data under the null hypothesis;(i.e. assuming that H0 is true)

When testing a mean, we know that the sample mean follows a normal distributionas long as we have n 25 or is known, so we can simply standardize our samplemean x.

test statistic: z= x n

or z =x

sn

, and z N(0, 1)

Step 3: Compute the p-value.

A p-value is the (conditional) probability of observing a test statistic that is asextreme or more extreme than what is actually observed given the null hypothesis istrue.

HA : < 0 HA : > 0 HA : =0p-value =P(Z < z) p-value =P(Z > z) p-value =P(|Z| > |z|)

Step 4: Make a decision.

p-value < Reject H0 p-value > Fail to reject H0

Step 5: State your conclusion in the context of the problem.

Reject H0: We have statistically significant evidence that... HA is true. Fail to reject H0: We do nothave statistically significant evidence that... HA is true.

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Example 2

(Section 6.2 Exercise 2) Heyde, Kuebrick, and Swanson measured the heights of 405 steelpunches of a particular type. These were all from a single manufacturer and were supposedto have heights of .500 in. (The stamping machine in which these are used is designed touse .500 in. punches.) The students measurements had x = .5002 in. and s = .0026 in.Use the five-step format and test the hypothesis that the mean height of such punches is

on spec (i.e. is .500 in.) Use = 0.1

Step 1: State the null and alternative hypotheses.

Step 2: Compute the test statistic and state its distribution.

Step 3: Compute the p-value.

Step 4: Make a decision.

Step 5: State your conclusion in the context of the problem.

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Example 3

(Section 6.2 Exercise 4) DuToit, Hansen, and Osborne measured the diameter of some no.10 machine screws with two different calipers. One student measured 50 screws using thedigital caliper and obtained x= 4.6858mm and s= 0.0290mm. Suppose that the nominaldiameter of the screws is 4.70 mm. Use the five-step format and test the hypothesis thatthe mean diameter is less than the nominal diameter. Use = .01.

Step 1: State the null and alternative hypotheses.

Step 2: Compute the test statistic and state its distribution.

Step 3: Compute the p-value.

Step 4: Make a decision.

Step 5: State your conclusion in the context of the problem.

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Hypothesis Testing Errors

Definition: AType I Error is rejecting H0 when H0 is true. P(type I error) = Definition: AType II Error is failing to reject H0 when H0 is false.

Hypothesis Testing as a Court Case

Innocent until proven guilty. She did it.

She didnt do it.

Collect and present evidence. Proof beyond a reasonable doubt. Conclusion

Guilty

Not Guilty

Hypothesis Testing and Confidence Intervals

Suppose we are considering a confidence interval with a confidence level of 95%.x 1.96 s

n, x + 1.96

sn

Also consider a hypothesis test of the form:

H0: = 0

HA : =0Then if0 is in the confidence interval,

x 1.96 sn

< 0 < x + 1.96 s

n

0 1.96 sn

< x < 0+ 1.96 s

n

1.96< x 0sn

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Then the p-value for the hypothesis test is P(|Z| > |z|)> .05.

Therefore, for any 0that is in a 95% confidence interval, the two sidedhypothesis test willnot be significant at the = 0.05 level of significance.

In general, for a (1 )100% confidence interval, and an -level hypothesis test for H0 := 0 vs. HA : =0, we will reject H0 if0 is outside the confidence interval and fail toreject H0 if0 is inside the confidence interval.

Example 4

Air bags were tested to determine the pressure present in the air bags 40 milliseconds afterreleasing the air bag. Suppose 50 bags were tested, the mean pressure is 6.5 psi and thestandard deviation is 0.25 psi.

Construct an 80% confidence interval for the mean.

Is there clear evidence that the mean pressure for all air bags under consideration isnot 6.5 psi?

Is there clear evidence that the mean pressure for all air bags under consideration isnot 6 psi?

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