Errors & Power

2 Results of Significant Test

1. P-value < alpha

2. P-value > alpha

Reject Ho & concludeHa in context

Fail to reject Ho & cannot conclude Ha

in context

Significance Level

needs to be stated before the data is producedneeds to be very small if Ho has been believed for yearsneeds to be small if the consequences are drastic if we reject it and we shouldn’t have

What if we’re wrong

Type I ErrorType II Error

These are not errors by people or the process; errors caused from variability and random chance

Type I Error

Reject Ho when it’s true

P(Type I Error) = significance level

“convicting and innocent person”

draw Normal curve picture of this

Type II Error

Failing 2 Reject Ho when it’s false

NOT accepting Ho, or saying it’s true

“let a guilty person go free”

The manager of a fast foot restaurant wants to reduce the proportion of drive through customers who have to wait more than two minutes. Based on store records, the proportion of customers who had to wait more than two minutes was p = 0.63. To reduce this proportion, the manage assigns an additional employee to assist with drive through orders. During the next month, the manager will collect a random sample of drive through times and test the following hypotheses:

Ho: p = 0.63Ha: p < 0.63

where p = the true proportion of drive through customers who have to wait more than two minutes for their food.

PROBLEM: Describe a Type I and Type II error in this setting and explain the consequence of each.

So why not just make our significance level very small so we don’t make a Type I

Error?

Your company markets a computerized device for detecting high blood

pressure. The device measures an individual’s blood pressure once per

hour at randomly selected times throughout a 12-hr period. Based on

sample results, the device determines whether there is significant evidence

that the individual’s actual mean systolic pressure is greater than 130. If

so, it recommends that person seek medical attention.

a) State Ho, Ha, and identify your parameter.

b) Describe Type I and Type II errors and their consequences

c) Which significance level would you choose: alpha = .01, .05, or .10

Type II ErrorFail 2 Reject the null when it is falseSo, we don’t know what the parameter is and we weren’t able to say it wasn’t what was stated in the null hypothesis.

POWER

probability that the test will reject the Ho at a given significance level when the specified alternative value is true (Ha)

P(Type II Error) = 1 - Power

Power & Planning

1. Setting a Significance Level - what are the consequences of a Type I error?

2. Practical Importance - how different does the sample need to be from the hypothesized value to warrant a change?

3. Sample Size - all errors and power are connected to sample size as well as the fact that the larger the sample the more time and money will be used

Remember the fast food restaurant's hypotheses:Ho: p = 0.63Ha: p < 0.63

where p = the true proportion of drive through customers who have to wait more than two minutes for their food.

1. Would the manager want to increase or decrease the significance level from 0.10?

2. For practical purposes, he decides that he needs to see the proportion drop to 0.53 to justify paying another employee.

3. Increase or decrease the sample size from 250?

draw Normal curve picture of this

What Goes Up Must Come Down

Lower significance level = Lower P(Type I Error)= Increase P(Type II Error)

orHigher sample size =

Increase Power = Decrease P(Type II Error)

You manufacture and sell a liquid product whose electrical conductivity is supposed to be 5. You plan to make 6 measurements of each lot of

product. If the product meets specifications, the mean of many measurements will be 5. You will therefore test

H0: mu = 5Ha mu doesn’t = 5

If the true conductivity is 5.1, the liquid is not suitable for its intended use. You learn that the power of your test at the 5% significance level

against the alternative mu = 5.1 is 0.23a) Explain what power = 0.23 means

b) Keeping alpha the same, could you increase your power by increasing or decreasing your sample size?

c) If alpha = 0.10 instead of 0.05, would the power increase or decrease?

d) Only change the alternative to 5.2, will your power increase or decrease?