Errors in Hypothesis Testing

Errors in Hypothesis Testing

​When making a decision in a hypothesis test, it is possible to make an ERROR:

There is a probability of making each error:

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Pr(Type I Error) = α​

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Pr(Type II Error) = β​

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(Note: 1 - Pr(Type II Error) = power of the test of the hypothesis)​

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​There is a trade-off between these two probabilities: it is not possible to make both probabilities small at the same time.

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For example,​ increasing α will result in a decrease in β.

​As noted, Pr(Type I Error) = α = level-of-significance

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Given α​, the critical/threshold value for the hypothesis test is found and this gives a rejection region (i.e the shaded region) and the non-rejection region (i.e. unshaded region).

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Both regions are then written in terms of the parameter estimate (e.g. the sample mean).

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The regions for the parameter estimate are used to find Pr(Type II Error) (and so the power of the test).​

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​You are a new employee at a marketing company. On your first day you are told that a fizzy-drink company has launched a new drink and believes that 70% of their customers will choose the new drink over the old drink. But, the fizzy-drink company would like you to test their belief. You conduct a market survey of 300 customers and find that 199 say they prefer the new drink and will switch.

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Answer the questions that follow.​

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​Example:

Multiple Choice

The hypotheses are:

• $$H_0:\ p\ =\ 0.7$$  

  $$H_1:\ p\ >0.7$$  

• $$H_0:\ p\ =\ 0.7$$  

  $$H_1:\ p\ \ne0.7$$  

• $$H_0:\ p\ <\ 0.7$$  

  $$H_1:\ p\ \ge0.7$$  

• $$H_0:\ p\ \ge0.7$$  

  $$H_1:\ p\ <\ 0.7$$  

Multiple Choice

The formula for the test-statistic is:

Fill in the Blank

What is the critical value for z if $$\alpha=5\%$$  ?

Multiple Choice

What is the criterion for NOT rejecting H₀ (express your answer in terms of the sample proportion)?

Fill in the Blank

What is the probability of a Type II Error if the true proportion is 0.60 (and not 0.7)? (Give your answer to 4 d.p.)

Fill in the Blank

What is the power of the test if the true proportion is 0.60?

(Give your answer to 4 d.p.)

​Instead of finding a critical value to compare to the calculated value of a test-statistic, the probability of obtaining a value for the test-statistic that is beyond the calculated value can be found.

The probability of a test-statistic value being beyond the calculated value is called a p-value (or, significance probability)​:

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Two-sided test: p = Pr(test-statistic < -calc) + Pr(test-statistic > calc)

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One-sided test​ (Negative): p = Pr(test-statistic < -calc)

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One-sided test​ (Positive): p = Pr(test-statistic > calc)

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​The decision for the hypothesis can then be made using a p-value:

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Decision:​ if p-value < α then Reject H₀

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Fill in the Blank

For the example question above, what is the p-value?

(Give your answer to 3 d.p.).

Multiple Choice

Referring to the p-value, should H₀ be rejected?

• Yes

• No

A food producer wants its factory to fill its cooking sauce bottles with 500ml of sauce. It is known that the standard deviation of the volume of cooking sauce in filled bottles is 50ml. In a spot check of the factory, 80 bottles are sampled and the sample mean amount of sauce in the bottles is found to be 510ml. Assume that the level of sauce in each bottle follows a normal model.

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Answer the questions that follow.​

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Multiple Choice

The hypotheses to test are...

• $$H_0:\ \mu\ =\ 500$$  

  $$H_1:\ \mu\ne500$$  

• $$H_0:\ \mu\ \ge500$$  

  $$H_1:\ \mu\ <\ 500$$  

• $$H_0:\ \mu\ \le500$$  

  $$H_1:\ \mu\ >\ 500$$  

Multiple Choice

What is the formula for the test-statistic?

Fill in the Blank

What is the critical value for test-statistic if $$\alpha=0.01$$  ?

(Give your answer to 3 d.p.)

Multiple Choice

What is the criterion for NOT rejecting H₀ (express your answer in terms of the sample mean)?

Fill in the Blank

What is the probability of a Type II Error if the true average volume of sauce is 480ml?

(Give your answer to 4 d.p.)

Fill in the Blank

What is the power of the test if the true average volume of sauce is 480ml?

(Give your answer to 4 d.p.)?

Fill in the Blank

What is the p-value for this hypothesis test?

(Give your answer to 3 d.p.)

Multiple Choice

Referring to the p-value, should H₀ be rejected?

• Yes

• No