Carrying out Significance TestsChapter 10

Section 2

What signs to use When writing Null and Alternate Hypothesis:

• (For our level we use: )

• For Null Hypothesis it is always • We always set the parameter of interest ________ to

what is _________ to be true

• For Alternate Hypothesis we use

• We set say the parameter is _____________ from what is ___________ to be true.

Which of these could be possible hypotheses for a significance test?

0: 0.5

: 0.5a

H p

H p

0: 0.5

: 0.5a

H p

H p

0: 0.5

: 0.5a

H p

H p

0: 0.5

: 0.5a

H p

H p

A B

C D

Write the Null and Alternate Hypothesis for the following

•You’ve been using the Random Digit Table all year, but one day you decided to check out its randomness. You count the number of times the digit “9” occurs in the last four rows of the table and find that there are only 12 “9”’s in the last 160 digits. You suspect that the table is flawed by having fewer “9”’s then it should and decided to do a significance test.

Write the Null and Alternate Hypothesis for the following

•The White House Press secretary comments that the president currently has a 72% favorable job approval rating. A pollster challenges this claim as being too high. His polling service has just conducted a random survey of 1000 people (calling both landline and cell phone numbers) and 660 people gave the president a favorable job approval rating. Do we have reason to doubt the press secretary?

Write the Null and Alternate Hypothesis for the following

•A 2009 nationwide random survey of 1500 adults asked the open-ended question, “What do you think is the most important problems facing this country today?” Sixty-nine percent responded with some form of economic problems (such as economy in general, unemployment/jobs, etc). Do these data provide good evidence that more than 2/3 of all adults believe that economic problems is the most important problem facing this country today?

Write the Null and Alternate Hypothesis for the following

• If a die is fair, then the number 5 should occur 1/6 of the time. You have a die that you suspect is loaded so that the number 5 lands face up more often then expected. You roll the die 200 times and get 44 5’s. Do we have evidence that the die is unfair (i.e. loaded)?

Write the Null and Alternate Hypothesis for the following

•According to the 2000 U.S. Census, 80.4% of all U.S. residents 25 years old or older have at least a high school diploma. In 2005, a random sample of 3,000 residents 25 years old or older found that 2,365 had at least a high school diploma. Is there evidence to suggest that the number of high school graduates is lower that reported in the Census?

Bad Physics Teacher?

•The head of the physics department has found over the years, students in the final exam average 70% with a standard deviation of 6.3%. One of his instructors seems to have a greater number of failures is his classes so the department head decides to do a hypothesis test. He takes SRS of 20 final exams from that teacher and finds that the average score is 67.3%. If there evidence at the 5% level that this teacher’s students have done worse that the general population of physics students?

P-values

•The p-value tells us the ______________ we would get _______ if m is the mean of the population.

•The p-value ______ ________tell us the probability of getting that result.

Choosing a Level Of Significance (alpha Level)

• How small a p-value is strong evidence against the Null Hypothesis?

• TWO MAIN THINGS TO CONSIDER…• How ____________ is H0? If Ho represents an

assumption that has been believed for years, __________ _____________will be needed to persuade

• What are the ________________ of rejecting H0? If rejecting H0 means that an ______________ changeover from one type of manufacturing to another strong evidence

Choosing a Level of Significance

• THERE IS NO SHARP BORDER BETWEEN “STATISTICALLY SIGNIFICANT” AND “STATISTICALLY INSIGNIFICANT”, ONLY INCREASINGLY STRONG EVIDENCE AS THE P-VALUE DECREASES

• ______ _________treat .05 as the magic level!!!

• A p = .051 or p = .049 does not provide strong evidence. A ____________ ___________ _________may be necessary to determine significance.

General Cautions• Look carefully at the __________ data, as well as the

__________ of a statistical test!

• Consider that there _______ _____an effect from a treatment even if a finding ________ statistically significant; and, vice versa, there ______ _______be an effect even if a finding is statistically significant.

• Design a study so that it has a high probability of finding a real effect.

General Cautions

•Design a study in such a way that the statistics used to analyze the study are ___________

•No “__________ in, ___________ out” data

•Make sure surveys or experiments are not ________ designed

•Understand that the use of probability in statistical inference is based on the assumption that our data is _________ ___________from the population of interest.

General Cautions• Be sure that all the data is represented in the

analysis so that “__________” don’t have an influence in the findings.

• Remember that a significance level of _____ means that you will get a _______________ finding one time out of 20 simply due to chance variation. If you run enough tests, you will get significant findings ________of the time even if nothing is going on.

•A confidence interval estimates the ______ of an effect rather than simply asking if it is too large to reasonably occur by chance alone.

Identify the null hypothesis, the alternative hypothesis, and test statistic

•The health of employees is monitored by periodically weighing them in. A sample of 54 employees has a mean weight of 183.9 lb. Assuming that σ is known to be 12.2 lb, use a

0.10 significance level to test the claim that the population mean of all such employees weights is less than 200 lb.

Identify the null hypothesis, the alternative hypothesis, and test statistic

•A random sample of 100 pumpkins is obtained and the mean circumference is found to be 40.5 cm. Assuming that the population standard deviation is known to be 1.6 cm, use a 0.05 significance level to test the claim that the mean circumference of all pumpkins is equal to 39.9 cm.

Example 1

•A company wishes to determine if the average salary of its clerks is less than the $340 believed by HR. The company researcher takes a sample of 64 clerks and finds that =$300 and s = $80. Test at 5% significance level.

Example 2

•A researcher claims that 10 year olds watch 6.6 hours of TV daily with a standard deviation of 2.5 hours. You try to verify this with the following sample data. A SRS of 100 10 year olds have a mean of 6.1 hours. Test the researcher claim at a 1% significance level.

Example 3

•An insurance company is reviewing its current policy rates. When originally setting the rates they believed that the average claim amount was $1,800. They are concerned that the true mean is actually higher than this, because they could potentially lose a lot of money. They randomly select 40 claims, and calculate a sample mean of $1,950. Assuming that the standard deviation of claims is $500, and set a= 05, test to see if the insurance company should be concerned.

Example

•A random number generator is supposed to produce random numbers that are uniformly distributed on the interval of 0 to 1. If this is true, the numbers generated come from a population with μ=0.5 and σ =0.2887. A command to generate 100 random numbers gives outcomes with mean =0.4365. Assume that that the population σ remains fixed. We want to test

• Ho: μ=0.5 Ha: μ≠0.5

• Is the result significant at the 5% level?

• Is the result significant at the 1% level?

x

Confidence Intervals and hypothesis Testing

•Confidence intervals can be used for 2 sided tests.

•You look for Ho to be in the _________.

• If it is you _________________the Ho.

• If not you __________________ the Ho.