1. Hypothesis Tests otherwise called Tests of Significance One Sample Means

2. Basketball Applet

3. Level of Significance Activity
Fish Oil vs Regular Oil

4. Hypothesis test will help me decide!
How can I tell if they really are underweight?
Let’s say a government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces).
Hypothesis test will help me decide!
Take a sample & find x.
But how do I know if this x is one that I expect to happen or is it one that is unlikely to happen?

5. What are hypothesis tests?
Calculations that tell us if a value occurs by random chance or not – if it is statistically significant
Is it . . .
a random occurrence due to variation?
a biased occurrence due to some other reason?

6. Nature of hypothesis tests -
How does a murder trial work?
Nature of hypothesis tests -
First - assume that the person is innocent
Then – must have sufficient evidence to prove guilty
First begin by supposing the claim or “effect” is NOT present
Next, see if data provides evidence against the supposition
Example: murder trial
Hmmmmm …
Hypothesis tests use the same process!

7. Another viewpoint in case you’re still not getting it
Someone/some business makes a pronouncement
You don’t believe it (if you did why test?)
The null hypothesis is their pronouncement
The alternative hypothesis is what you really think is going on
So hypothesis tests are probability tests that measure how well the data and the pronouncement agree
Hypothesis tests DO NOT make decisions regarding the trueness of the alternative hypothesis

8. Steps: Assumptions Hypothesis statements & define parameters
Notice the steps are the same except we add hypothesis statements – which you will learn today
Assumptions
Hypothesis statements & define parameters
Calculations
Conclusion, in context

9. Assumptions for z-test (t-test):
YEA –
These are the same assumptions as confidence intervals!!
Have an SRS of context
Independent Sample (Pop > 10n)
Distribution is (approximately) normal
Given
Large sample size
s is known

10. Writing Hypothesis statements:
Null hypothesis – is the statement being tested; this is a statement of “no effect” or “no difference” - is an equality statement
In other words, the test that everything is as it should be
H0:

11. Writing Hypothesis statements:
Alternative hypothesis – is the statement that we suspect is true
In other words, we think that the original claim is not true, and we think that the actual results are different in some way
Ha:

12. The form: Null hypothesis H0: parameter = hypothesized value
H0 MUST be “=“ !
Null hypothesis
H0: parameter = hypothesized value
Alternative hypothesis
Ha: parameter > hypothesized value
Ha: parameter < hypothesized value
Ha: parameter = hypothesized value

13. Example 2: A government agency has received numerous complaints that a particular restaurant has been selling underweight hamburgers. The restaurant advertises that it’s patties are “a quarter pound” (4 ounces).
State the hypotheses :
Must define what µ is
H0: m = 4
Ha: m < 4
Where m is the true mean weight of hamburger patties

14. Example 3: A car dealer advertises that is new subcompact models get 47 mpg. You suspect the mileage might be overrated.
State the hypotheses :
Must define what µ is
H0: m = 47
Ha: m < 47
Where m is the true mean mpg

15. Example 4: Many older homes have electrical systems that use fuses rather than circuit breakers. A manufacturer of 40-A fuses wants to make sure that the mean amperage at which its fuses burn out is in fact 40. If the mean amperage is lower than 40, customers will complain because the fuses require replacement too often. If the amperage is higher than 40, the manufacturer might be liable for damage to an electrical system due to fuse malfunction. State the hypotheses :
H0: m = 40
Ha: m = 40
Where m is the true mean amperage of the fuses

16. Activity: For each pair of hypotheses, indicate which are not legitimate & explain why
Must use parameter (population)
x is a statistics (sample)
Must be NOT equal!
p is the population proportion!
Must use same number as H0!
r is parameter for population correlation coefficient – but H0 MUST be “=“ !

17. In other words . . . is it far out in the tails of the distribution?
P-values -
The probability that the test statistic would have a value as extreme or more than what is actually observed
In other words is it far out in the tails of the distribution?

18. P-values -
The smaller the p-value, the stronger the evidence against H0 provided by the data
Large p-values fail to give evidence against H0

19. Level of significance -
This is the amount of evidence necessary before we begin to doubt that the null hypothesis is true
Is the probability that we will reject the null hypothesis, assuming that it is true
Denoted by a
Can be any value
Usual values: 0.1, 0.05, 0.01
Most common is 0.05

20. Statistically significant –
The p-value is as small or smaller than the level of significance (a)
If p > a, “fail to reject” the null hypothesis at the a level.
If p < a, “reject” the null hypothesis at the a level.

21. Facts about p-values: ALWAYS make decision about the null hypothesis!
Large p-values show support for the null hypothesis, but never that it is true!
Small p-values show support that the null is not true.
Double the p-value for two-tail (=) tests
Never accept the null hypothesis!

22. In AP Statistics, which is what colleges accept as a college level course, they insist that you Never “accept” the null hypothesis or say that it is “true”!
However, this is not always followed. For example, IB allows students to “accept” the null hypothesis, or say that it is “true”.
You need to be aware of both mentalities

23. At an a level of .05, would you reject or fail to reject H0 for the given p-values?
.03
.15
.45
.023
Reject
Fail to reject
Fail to reject
Reject

24. Calculating p-values
With z-test statistic (same as z-score but for statistic (ie sample))
normalcdf(lb, ub,[mean,standard deviation])
You may have to find the z-test number first

25. Draw & shade a curve & calculate the p-value:
1) right-tail test z = 1.6; n = 20
2) left-tail test z = -2.4; n = 15
3) two-tail test z = 2.3; n = 25

26. Writing Conclusions:
A statement of the decision being made (reject or fail to reject H0) & why (linkage)
A statement of the results in context. (state in terms of Ha)
AND

27. Be sure to write Ha in context (words)!
“Since the p-value < (>) a, I reject (fail to reject) the H0. There is (is not) sufficient evidence to suggest that Ha.”
Be sure to write Ha in context (words)!

28. Example 5: Drinking water is considered unsafe if the mean concentration of lead is 15 ppb (parts per billion) or greater. Suppose a community randomly selects of 25 water samples and computes a t-test statistic of Assume that lead concentrations are normally distributed. Write the hypotheses, calculate the p-value & write the appropriate conclusion for a = 0.05.
H0: m = 15
Ha: m > 15
Where m is the true mean concentration of lead in drinking water
P-value = normcdf(2.1,10^99) =.0179
z=2.1
Since the p-value < a, I reject H0. There is sufficient evidence to suggest that the mean concentration of lead in drinking water is greater than 15 ppb.

29. Example 6: A certain type of frozen dinners states that the dinner contains 240 calories. A random sample of 12 of these frozen dinners was selected from production to see if the caloric content was greater than stated on the box. The t-test statistic was calculated to be 1.9. Assume calories vary normally. Write the hypotheses, calculate the p-value & write the appropriate conclusion for a = 0.05.
H0: m = 240
Ha: m > 240
Where m is the true mean caloric content of the frozen dinners
P-value = normcdf(1.9,10^99) =.0287
z=1.9
Since the p-value < a, I reject H0. There is sufficient evidence to suggest that the true mean caloric content of these frozen dinners is greater than 240 calories.

30. Formulas:
s known: m
z =

31. Example 7: The Fritzi Cheese Company buys milk from several suppliers as the essential raw material for its cheese. Fritzi 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 degrees and a standard deviation of Added water raises the freezing temperature toward 0 degrees, the freezing point of water (in Celsius). The laboratory manager measures the freezing temperature of five randomly selected lots of milk from one producer with a mean of degrees. Is there sufficient evidence to suggest that this producer is adding water to his milk?

32. I have an SRS of milk from one producer Normal? How do you know?
Assumptions:
Independent sample?
I have an SRS of milk from one producer
Normal?
How do you know?
Assume more than 50 lots of milk
The freezing temperature of milk is a normal distribution. (given)
Do you know s?
s is known
What are your hypothesis statements? Is there a key word?
H0: m =
Ha: m >
where m is the true mean freezing temperature of milk
Plug values into formula.
p-value = normalcdf(1.9566,1E99)=.0252
Use normalcdf to calculate p-value.
a = .05

33. Compare your p-value to a & make decision Conclusion:
Since p-value < a, I reject the null hypothesis.
There is sufficient evidence to suggest that the true mean freezing temperature is greater than This suggests that the producer is adding water to the milk.
Write conclusion in context in terms of Ha.

34. Example 9: The Wall Street Journal (January 27, 1994) reported that based on sales in a chain of Midwestern grocery stores, President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week with a SD of Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $ Does this indicate that the sales of the cookies is different from the earlier figure?

35. Assume:
Have an SRS of weeks
Assume one week’s sale is independent of the next
Distribution of sales is approximately normal due to large sample size
s unknown
H0: m = where m is the true mean cookie sales per
Ha: m ≠ week
Since p-value < a of 0.05, I reject the null hypothesis. There is sufficient evidence to suggest that the sales of cookies are different from the earlier figure.

36. Example 9 Cont.: President’s Choice Chocolate Chip Cookies were selling at a mean rate of $1323 per week and a SD of $275. Suppose a random sample of 30 weeks in 1995 in the same stores showed that the cookies were selling at the average rate of $1208. Compute a 95% confidence interval for the mean weekly sales rate.
CI = ($1109.6, $1306.4)
Based on this interval, is the mean weekly sales rate statistically different from the reported $1323?

37. What do you notice about the decision from the confidence interval & the hypothesis test?
You expect that the significance test would support the confidence interval and visa versa. That means that if you reject the null hypothesis, you would expect that the null hypothesis value to be outside of your calculated interval.

38. A special type of t-inference
Matched Pairs Test
A special type of t-inference

39. Matched Pairs – two forms
Pair individuals by certain characteristics
Randomly select treatment for individual A
Individual B is assigned to other treatment
Assignment of B is dependent on assignment of A
Individual persons or items receive both treatments
Order of treatments are randomly assigned or before & after measurements are taken
The two measures are dependent on the individual

40. Is this an example of matched pairs?
1)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 from 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
No, there is no pairing of individuals, you have two independent samples

41. Is this an example of matched pairs?
2) 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
No, there is no pairing of individuals, you have two independent samples – If you would have the same people taste both brands in random order, then it would bean example of matched pairs.

42. Is this an example of matched pairs?
3) 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.
Yes, you have two measurements that are dependent on each individual.

43. 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 collected the following data on 15 randomly selected days over the past month. (Note: days were not consecutive.)
You may subtract either way – just be careful when writing Ha
Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Morning
After-noon
Since you have two values for each day, they are dependent on the day – making this data matched pairs
First, you must find the differences for each day.

44. -1 -2 I subtracted: Morning – afternoon
Day 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Morning
After-noon
Differences -1 -2
I subtracted:
Morning – afternoon
You could subtract the other way!
Assumptions:
Have an SRS of days for whale-watching
s of difference is 1.6
Since the normal probability plot is approximately linear, the distribution of difference is approximately normal.
You need to state assumptions using the differences!

45. Differences -1 -2 1 2
Is there sufficient evidence that more whales are sighted in the afternoon?
Be careful writing your Ha!
Think about how you subtracted: M-A
If afternoon is more should the differences be + or -?
Don’t look at numbers!!!!
If you subtract afternoon – morning; then Ha: mD>0
H0: mD = 0
Ha: mD < 0
Where mD is the true mean difference in whale sightings from morning minus afternoon
Notice we used mD for differences
& it equals 0 since the null should be that there is NO difference.

46. finishing the hypothesis test:
Differences -1 -2 1 2
finishing the hypothesis test:
Since p-value > a, I fail to reject H0. There is insufficient evidence to suggest that more whales are sighted in the afternoon than in the morning.
In your calculator, perform a t-test using the differences (L3)
Notice that if you subtracted A-M, then your test statistic
t = , but p-value would be the same