Test of Significance (Hypothesis Tests) Notes
Hypothesis Tests Notes
1. A lottery advertises that 10% of people who buy a lottery ticket win a prize. Recently, the organization that oversees this lottery has received several complaints claiming that there are fewer winners than there should be.
What do hypothesis tests answer?
The Idea:
Steps for a Hypothesis Test:
1)
2)
3)
4)
Conditions for a Proportions z-Test:
Conditions for a Means t-Test:
2. Bottles of a popular cola are supposed to contain 300 mL of cola. There is some variation from bottle to bottle. An inspector, who suspects that the bottler is under-filling, measures the contents of six randomly selected bottles. Are the conditions met?
299.4 297.7 298.9 300.2 297 301
Writing Hypothesis Statements
Null hypothesis:
Alternative hypothesis:
How they look:
3. A lottery advertises that 10% of people who buy a lottery ticket win a prize. Recently, the organization that oversees this lottery has received several complaints claiming that there are fewer winners than there should be. State the hypotheses we'd use to test a sample of lottery tickets.
4. A consumer magazine advertizes a new compact car as getting, on average, 22 mpg. A dealership believes this ad underrates the car's mileage. State the hypotheses we'd use to test a sample of compact cars.
5. The carbon dioxide (CO2) level in a home varies greatly, but a typical level is around .06%. Since CO2 concentration outdoors is typically lower, an indoor level of less than .06% may indicate that the home is not properly sealed. Indoor CO2 levels above .06%, on the other hand, may cause residents to feel drowsy and report that the air feels poor. State the hypotheses we'd use to test CO2 levels in a sample of homes.
The Golden Rules of Hypotheses:
Are these valid hypotheses? If not, why?
a) H0: ( = 15; Ha: ( = 15
b) H0: [pic]= 123; Ha: [pic] < 123
c) H0: p = 0.1; Ha: p [pic] 0.1
d) H0: ( = .4; Ha: ( > .6
e) H0: p ≠ 0; Ha: p = 0
P-value:
Level of Significance:
Statistically Significant:
( If p-value < (,
( If p-value > (,
Golden Rules of p-Values:
At an ( level of .05, would you reject or fail to reject the H0?
a) .03 b) .15
c) .45 d) .023
Calculating p-Values:
• For a z-test:
• For a t-test:
Draw & shade a curve, and calculate the p-value:
1) right-tail test ( z = 1.6
2) left-tail test ( z = -2.4
3) two-tailed test ( z = 2.3
Hypothesis Test Conclusions:
6. To be considered two percent milk, a carton of milk must have at most a 2.5% fat concentration. A consumer randomly selects 25 two percent milk cartons and computes a z-test statistic of 2.1. Write the hypotheses, calculate the p-value, and write the appropriate conclusion for α = .05.
7. A lottery advertises that 10% of people who buy a lottery ticket win a prize. Recently, the organization that oversees this lottery has received several complaints claiming that there are fewer winners than there should be. A group of citizens collects a random sample of lottery tickets and finds a test statistic of -1.35. Assume the conditions are met. Write the hypotheses, calculate the p-value, and write the appropriate conclusion for α = 0.05.
Test Statistic for a Proportion:
8. A company is willing to renew its advertising contract with a local radio station only if the station can prove that more than 20% of the residents of the city have heard the ad. The radio station conducts a random sample of 400 people and finds that 90 have heard the ad. Is this sufficient evidence for the company to renew its contract?
9. A supernatural magazine claims that 63% of Americans believe in ghosts. Gallup surveys 200 randomly selected Americans and finds that 54% of them say they believe in ghosts. At the 1% significance level, does the Gallup poll give us evidence to doubt the magazine's claim?
Test Statistic for a Mean:
10. Kraft buys milk from several suppliers as the essential raw material for its cheese. Kraft suspects that some producers are adding water to their milk to increase their profits. Excess water can be detected by determining the freezing point of milk. The freezing temperature of natural milk varies normally, with a mean of -0.545 degrees and a standard deviation of 0.008. Added water raises the freezing temperature toward 0 degrees, the freezing point of water (in Celsius). A laboratory manager measures the freezing temperature of five randomly selected lots of milk from one produce and finds a mean of -0.538 degrees. Is there sufficient evidence to suggest that this producer is adding water to the milk?
11. The Degree of Reading Power (DRP) is a test of the reading ability of children. Here are DRP scores for a random sample of 44 third-grade students in a suburban district:
40 26 39 14 42 18 25
43 46 27 19 47 19 26
35 34 15 44 40 38 31
46 52 25 35 35 33 29
34 41 49 28 52 47 35
48 22 33 41 51 27 14
54 45
At the .1 significance level, is there sufficient evidence to suggest that this district’s third graders’ reading ability is different than the national average of 34?
12. a) In 2011, Mrs. Field’s chocolate chip cookies were selling at a mean rate of $1323 per week. A random sample of 30 weeks in 2012 in the same stores showed that the cookies were selling at an average rate of $1228 with standard deviation $275. Compute a 95% confidence interval for the mean weekly sales rate.
Based on this interval, is the mean weekly sales rate statistically lower than the 2011 figure?
b) Does the data indicate that the sales of the cookies in 2012 were lower than the 2011 figure?
Matched Pairs Test
Two Types:
Is this matched pairs?
a) A college wants to see if there’s a difference in time it took last year’s class to find a job after graduation and the time it took the class form five years ago to find work after graduation. Researchers take a random sample from both classes and measure the number of days between graduation and first day of employment
b) A pharmaceutical company wants to test its new weight-loss drug. Before giving the drug to a random sample, company researchers take a weight measurement on each person. After a month of using the drug, each person’s weight is measured again.
c) In a taste test, a researcher asks people in a random sample to taste a certain brand of spring water and rate it. Another random sample of people is asked to taste a different brand of water and rate it. The researcher wants to compare these samples.
13. A whale-watching company noticed that many customers wanted to know whether it was better to book an excursion in the morning or the afternoon. To test this question, the company recorded the number of whales spotted in the morning and afternoon on 15 randomly selected days over the past month.
Day |1 |2 |3 |4 |5 |6 |7 |8 |9 |10 |11 |12 |13 |14 |15 | |Mor-ning |8 |9 |7 |9 |10 |13 |10 |8 |2 |5 |7 |7 |6 |8 |7 | |After-noon |8 |10 |9 |8 |9 |11 |8 |10 |4 |7 |8 |9 |6 |6 |9 | |
| | | | | | | | | | | | | | | | |
a) Is there sufficient evidence that more whales are sighted in the afternoon?
b) Construct a 90% confidence interval for the mean difference in whale sightings. Does your conclusion match the conclusion from the hypothesis test?
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Since the p-value < (>) (, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha.
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