Testing a ClaimI’m a great free-throw shooter!

Significance Tests

A significance test is a formal procedure for comparing observed data with a claim (also called a hypothesis) whose truth we want to assess.

I’m a Great Free-Throw Shooter!

Our virtual basketball player in the previous activity claimed to be an 80% free-throw shooter. Suppose that he shoots 50 free throws and makes 32 of them. His sample proportion of made shots is

64.050

32ˆ p

80.0:0 pH Null Hypothesis

80.0: pH a Alternate Hypothesis

One-sided

3/400= 0.0075

P-Value

DEFINITION: P-value

The probability, computed assuming H0 is true, that the statistic (such as or X) would take a value as extreme as or more extreme than the one actually observed is called the P-value of the test. The smaller the P-value, the stronger the evidence against H0 provided by the data.

p̂

Studying Job Satisfaction

Does the job satisfaction of assembly-line workers differ when their work is machine-paced rather than self-paced? One study chose 18 subjects at random from a company with over 200 workers who assembled electronic devices. Half of the workers were assigned at random to each of two groups. Both groups did similar assembly work, but one group was allowed to pace themselves while the other group used an assembly line that moved at a fixed pace. After two weeks, all the workers took a test of job satisfaction. Then they switched work setups and took the test again after two more weeks. The response variable is the difference in satisfaction scores, self-paced minus machine-paced.

(a) Describe the parameter of interest in this setting.

(a) The parameter of interest is the mean μ of the differences (self-paced minus machine-paced) in job satisfaction scores in the population of all assembly-line workers at this company.

(b) State appropriate hypotheses for performing a significance test.

0:

0:0

aH

HTwo sided

For each of the following settings, (a) describe the parameter of interest, and (b) state appropriate hypotheses for a significance test.

According to the Web site sleepdeprivation.com, 85% of teens are getting less than eight hours of sleep a night. Jannie wonders whether this result holds in her large high school. She asks an SRS of 100 students at the school how much sleep they get on a typical night. In all, 75 of the responders said less than 8 hours.

(a) p = proportion of students at Jannie’s high school who get less than 8 hours of sleep at night.

(b) H0: p = 0.85 and Ha: p ≠ 0.85.

As part of its 2010 census marketing campaign, the U.S. Census Bureau advertised “10 questions, 10 minutes—that’s all it takes.” On the census form itself, we read, “The U.S. Census Bureau estimates that, for the average household, this form will take about 10 minutes to complete, including the time for reviewing the instructions and answers.” We suspect that the actual time it takes to complete the form may be longer than advertised.

(a) μ = mean amount of time that it takes to complete the census form.

(b) H0: μ = 10 and Ha: μ > 10

For the job satisfaction study described, the hypotheses are

where μ is the mean difference in job satisfaction scores (self-paced − machine-paced) in the population of assembly-line workers at the company. Data from the 18 workers gave and sx = 60. That is, these workers rated the self-paced environment, on average, 17 points higher.

0:

0:0

aH

H

17x

Researchers performed a significance test using the sample data and obtained a P-value of 0.2302

(a) Explain what it means for the null hypothesis to be true in this setting.

(b) Interpret the P-value in context.an average difference of 17 or more points between the two work environments would happen 23% of the time just by chance in random samples of 18 assembly-line workers when the true population mean is μ = 0

(c) Do the data provide convincing evidence against the null hypothesis? Explain.

In a nutshell, our conclusion in a significance test comes down to:

P-Value small Reject H0 Conclude Ha in context

P-Value large Fail to reject H0 Cannot conclude Ha in context

How small of a P-value is small? Smaller than an α = 0.05 or 0.01

P-Value < α Reject H0 Conclude Ha in context

P-Value ≥ α Fail to reject H0 Cannot conclude Ha in context

Better BatteriesA company has developed a new deluxe AAA battery that is supposed to last longer than its regular AAA battery. However, these new batteries are more expensive to produce, so the company would like to be convinced that they really do last longer. Based on years of experience, the company knows that its regular AAA batteries last for 30 hours of continuous use, on average. The company selects an SRS of 15 new batteries and uses them continuously until they are completely drained. A significance test is performed using the hypotheses

hours 30:

hours 30:0

aH

H

where μ is the true mean lifetime of the new deluxe AAA batteries. The resulting P-value is 0.0276.

What conclusion would you make for α=0.05 significance levels? For α=0.01?

(a) Since the P-value, 0.0276, is less than α = 0.05, the sample result is statistically significant at the 5% level. We have sufficient evidence to reject H0 and conclude that the company’s deluxe AAA batteries last longer than 30 hours, on average.

Answer:

(b) Since the P-value, 0.0276, is greater than α = 0.01, the sample result is not statistically significant at the 1% level. We do not have enough evidence to reject H0 in this case. Therefore, we cannot conclude that the deluxe AAA batteries last longer than 30 hours, on average.

Tasty ChipsFor his second semester project in AP Statistics, Zenon decided to investigate if students at his school prefer name-brand potato chips to generic potato chips. He randomly selected 50 students and had each student try both types of chips, in random order. Overall, 34 of the 50 students preferred the name-brand chips. Zenon performed a significance test using the hypotheses: H0 : p = 0.5

Ha: p > 0.5where p = the true proportion of students at his school that prefer name-brand chips. The resulting P-value was 0.0055.

Problem: What conclusion would you make at each of the following significance levels?(a) = 0.01 (b) = 0.001

Type I and Type II Errors

If we reject H0 when H0 is true, we have committed a Type I error

If we fail to reject H0 when H0 is false, we have committed a Type II error.

Perfect PotatoesA potato chip producer and its main supplier agree that each shipment of potatoes must meet certain quality standards. If the producer determines that more than 8% of the potatoes in the shipment have “blemishes,” the truck will be sent away to get another load of potatoes from the supplier. Otherwise, the entire truckload will be used to make potato chips. To make the decision, a supervisor will inspect a random sample of potatoes from the shipment. The producer will then perform a significance test using the hypotheses H0: p=0.08

Ha: p>0.08

Where p is the actual proportion of potatoes with blemishes in a given truckload.

Describe a Type I and a Type II error in this setting, and explain the consequences of each.

Faster fast food?The manager of a fast-food restaurant want to reduce the proportion of drive-through customers who have to wait more than 2 minutes to receive their food once their order is placed. Based on store records, the proportion of customers who had to wait at least 2 minutes was p = 0.63. To reduce this proportion, the manager 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: H0: p=0.63

Ha: p<0.63where p = the true proportion of drive-through customers who have to wait more than 2 minutes after their order is placed to receive their food.

Describe a Type I and a Type II error in this setting and explain the consequences of each.

Significance and Type I Error

P(type I error) = Significance level (α)

Power and Type II Error

P(type II error) = β

Power =1 - β

DEFINITION: PowerThe power of a test against a specific alternative is the probability that the test will reject H0 at a chosen significance level α when the specified alternative value of the parameter is true.

Lower α Higher n

Higher n Higher power

Exercises on page 546, # 1 – 15 odds,# 19, 21,23, 25

Read Section 9.2