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DO NOW

´ Find your number in the lists posted in the bulletin boards (back wall and main entrance).

´ Find the seat with your number and that will be your seat from now on.

´ SILENCE YOUR PHONE and put it in the pocket that has your number in the bulletin board (back wall).

´ NO EXCEPTIONS and NO PHONES in backpack. If I see your phone, I will take it!!!

´ No food or drinks (except for water) are allowed in my room. Finish your food outside before you enter my classroom.

Testing the Difference Between Two Means: 𝜎 is knownUsing the z Test

Comparing Two Samples

´ Up to now, we have looked at inference for one sample of continuous data

´ Our next focus in this course is comparing the data from two different samples

• For now, we will assume that these two different samples are independent of each other and come from two distinct populations.

Population 1:µ1 , s1

Sample 1: , s1

Population 2: µ2 , s2

Sample 2: , s2

Blackout Baby Boom Revisited

´ Nine months (Monday, August 8th) after Nov 1965 blackout, NY Times claimed an increased birth rate

´ Already looked at single two-week sample: found no significant difference from usual rate (430 births/day)

´ What if we instead look at difference between a sample of July and August in 1966 and a sample of July and August in 1965?

´ How about weekends vs weekdays?

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Remember the steps: 1. Hypothesis Remember the steps: 1. Hypothesis

Remember the steps: 2. Critical Values

´When the population standard deviation (𝜎)is known we use 𝑧-scores.

Remember the steps: 2. Critical Values

´ If you do not remember or have limited access to the critical values for the 3 most common levels of significances, you can always use your 𝑧-table for any level.

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Remember the steps: 3. Test Values

´ Since the standard deviation (𝜎)is known, our test value will be 𝑧.

´ Remember the one for one-sample hypothesis test?

Remember the steps: 3. Test Values

´ For a two-sample hypothesis test, we need to test the observed difference and the expected difference.

Remember the steps: 4. Decision

´ Reject the null of the comparison between the test value and the critical value follows the alternative hypothesis.

´ Otherwise, do not reject!

Remember the steps: 5. Conclusion

´ Always follow the claim.´ If the claim is in the null…

´When you reject, then there is enough evidence to reject the claim that …

´When you can’t reject, then there is not enough evidence to reject the claim that …

´ If the claim is in the alternative…´When you reject, then there is enough evidence to support the

claim that …´When you can’t reject, then there is not enough evidence to

support the claim that …

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Learning-by-doing: Example 1 Learning-by-doing: Example 1

Learning-by-doing: Example 1 Learning-by-doing: Example 1

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Learning-by-doing: Example 2 Learning-by-doing: Example 2

Consider your options!

´You can compare test values with critical values.´You can find the p-value for the critical value and

compare with the level of significance. ´If it is less than alpha, you reject!

Partner/Independent Work

´ Question #4 will require you to go to https://goo.gl/cvr7WNand manipulate the data in the spreadsheet.

´ Use your Chromebook or personal computer. ´ Add a worksheet, copy the data and paste it in your own

workspace.

DO NOW

´ Find your number in the lists posted in the bulletin boards (back wall and main entrance).

´ Find the seat with your number and that will be your seat from now on.

´ SILENCE YOUR PHONE and put it in the pocket that has your number in the bulletin board (back wall).

´ NO EXCEPTIONS and NO PHONES in backpack. If I see your phone, I will take it!!!

´ No food or drinks (except for water) are allowed in my room. Finish your food outside before you enter my classroom.

Testing the Difference Between Two Means: 𝜎 is knownUsing the 𝑡 Test

Tale of two Mexican restaurant: La Pasadita vs. La Tienda´There is a rumor that the waiting time for your food at

La Pasadita (once the order has been placed) is greater than the waiting time at La Tienda. The data shown below is in minutes and both are normally distributed. Use a level of significance of 0.05.

La Pasadita La Tienda20.4 20.224.2 16.915.4 18.521.4 17.320.2 20.518.521.5

Tale of two Mexican restaurant: La Pasadita vs. La Tienda´Step 1: Hypothesis

´ 𝐻$: 𝜇' − 𝜇) = 0 and 𝐻,: 𝜇' − 𝜇) > 0 [claim]

´Step 2: Critical Value´ Since the test is two-tailed, since 𝛼 =0.05, and since the variances (and sample

size) are unequal, the degrees of freedom are the smaller of the two samples. La Pasadita has seven observations and La Tienda has five.

´ d.f. =  5  –  1   =  4

´ C.V = +2.132

Tale of two Mexican restaurant: La Pasadita vs. La Tienda´We need info on mean and standard deviation…

´Step 3: Test Value (t-test)

´𝑡 = 1.12

Mean Standard DeviationLa Pasadita (n = 7) 20.23 2.74La Tienda (n = 5) 18.68 1.64

Tale of two Mexican restaurant: La Pasadita vs. La Tienda´Step 4: Decision

´Remember our C.V = +2.132´To  reject  the  null  we  want  to  be  more  than  2.132 because  it  is  a  one-­‐‑tailed  test.  

´𝑡 = 1.12 is NOT more than +2.132.´We cannot reject the null hypothesis.

´Step 5: Conclusion´Since the claim is in the alternative and we couldn’t reject,

then there is not enough evidence to support the claim that he time you wait for the food once the order has been placed in La Pasadita is more than in La Tienda.

Tale of two Mexican restaurant: La Pasadita vs. La Tienda´Analyzing the output

´ Since the p-value is 0.289, i.e. greater than 0.05 (or 5 percent), it can be concluded that there is no difference between the means.

´ If we are wrong, then we are committing a type II error. However, that p-value is way off 0.05.

´ If we reject the null, we are taking a 28.9 percent risk of being wrong and commit a type I error.

Learning-by-doing: Your turn #1

Learning-by-doing: Your turn #1

´Step 1: Hypothesis´𝐻$: 𝜇' − 𝜇) = 0 and 𝐻,: 𝜇' − 𝜇) ≠ 0 [claim]

Learning-by-doing: Your turn #1

´Step 2: Critical Value´ Find the critical values. Since the test is two-tailed, since my alpha is

0.05, and since the variances are unequal, the degrees of freedom are the smaller of the two. In this case, the degrees of freedom are 7. Hence, from the t-table, the critical values are -2.365 and 2.365.

Learning-by-doing: Your turn #1

´Step 3: Compute the test value.

Learning-by-doing: Your turn #1

´ Step 4: Decision´ Remember our C.V = ±2.365

´To  reject  we  want  to  be  less  than  −2.365  or  more  than  2.365 because  it  is  a  two-­‐‑tailed  test.  ´ 𝑡 = −0.57 is NOT less than −2.365.´ We cannot reject the null hypothesis.

Learning-by-doing: Your turn #1

´Step 5: Conclusion´There is not enough evidence to support the claim that the

average size of the farms is different.

Learning-by-doing: Your turn #2Moisture content (by percent) for random samples of different fruits and

vegetables

At the 0.05 level of significance, can it be concluded that fruits differ from vegetables in average moisture content?

Learning-by-doing: Your turn #2Moisture content (by percent) for random samples of different fruits and

vegetables

Consider your options!

´You can compare test values with critical values.´You can find the p-value for the critical value and

compare with the level of significance. ´If it is less than alpha, you reject!

Partner/Independent Work

´ Work on your worksheet. ´ Do a couple with a partner and at least one by yourself. ´ Show it to me when you are done.

2/7/18

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DO NOW

´ Find your number in the lists posted in the bulletin boards (back wall and main entrance).

´ Find the seat with your number and that will be your seat from now on.

´ SILENCE YOUR PHONE and put it in the pocket that has your number in the bulletin board (back wall).

´ NO EXCEPTIONS and NO PHONES in backpack. If I see your phone, I will take it!!!

´ You will need your CHROMEBOOK.´ No food or drinks (except for water) are allowed in my room.

Finish your food outside before you enter my classroom.

Testing the Difference Between Two Means:Dependent SamplesUsing the 𝑡 Test

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Paired-Samples (dependent samples) t-tests´2 samples are statistically related

´Less affected by individual differences´reduces variance due to error

´Repeated-measures´2 measurements on same individual

´Matched-subjects´Match pairs on some variable(s)´Split pairs into 2 groups ~

Dependent Samples

´Before and After Measurements on the same individual

´Studies of twins´Studies of spouses´Pre- and Post Tests

Individual

1

2

3

4

5

6

Before

32

11

21

17

30

38

After

39

15

35

13

41

39

d

-7

-4

-14

4

-11

-1

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How do we perform the test?

1. Hypothesis

2. Find the critical value using 𝑡-table.Ø You will need the d.f.Ø Match the number in that row with the significance level.

How do we perform the test?

3. Find the test value (using spreadsheet)a) Find the differences of the values of the pairs of data.b) Find the mean of those differences (𝐷#). c) Find the standard deviation of those difference (𝑠%).d) Then, use formula:

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How do we perform the test?

4. DecisionØ For right-tailed: Reject if the test value is greater than critical

value.Ø For left-tailed: Reject if the test value is less than the critical

value. Ø For two-tailed: Reject if the test-value is greater or less than the

positive and negative critical values, respectively. Ø For p-values: Reject if p-value is less than alpha.

5. ConclusionØ Focus on where the claim is.Ø Enough evidence to reject the claim…Ø Enough evidence to support the claim…

Learning-by-doing: Your turn #1

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Learning-by-doing: Your turn #1

´Step 1: Hypothesis´𝐻': 𝜇% = 0 and 𝐻,: 𝜇% < 0 [claim]

Learning-by-doing: Your turn #1

´Step 2: Critical Value´Find the critical value. The degrees of freedom are𝑛 − 1. In this

case, d.f. = 8 − 1 = 7. The critical value for a left-tailed test with a significance level of 0.05 is −1.895.

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Learning-by-doing: Your turn #1´Step 3: Compute the test value. Enter data.

Learning-by-doing: Your turn #1´Step 3: Compute the test value. Calculate differences.

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Learning-by-doing: Your turn #1´Step 3: Compute the test value.

• 𝐷# = −2.375• 𝑠% = 4.84

´.

Learning-by-doing: Your turn #1

´Step 4: Decision´Remember our C.V = −1.895

´Torejectwewanttobelessthan−1.895becauseitisaone(left)-tailedtest.

´𝑡 = −1.388 is NOT less than −1.895.´The decision is not to reject the null hypothesis at 𝛼 = 0.05.

´Step 5: Conclusion´ There is not enough evidence to support the claim that the

vitamin increases the strength of weight lifters.

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Learning-by-doing: Your turn #2

Learning-by-doing: Your turn #2

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Learning-by-doing: Your turn #2

Learning-by-doing: Your turn #3

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Learning-by-doing: Your turn #3

Learning-by-doing: Your turn #3

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Partner/Independent Work

´ Work on your worksheet. ´ Do a couple with a partner and at least one by yourself. ´ Show it to me when you are done.

2/11/18

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DO NOW

´ Find your number in the lists posted in the bulletin boards (back wall and main entrance).

´ Find the seat with your number and that will be your seat from now on.

´ SILENCE YOUR PHONE and put it in the pocket that has your number in the bulletin board (back wall).

´ NO EXCEPTIONS and NO PHONES in backpack. If I see your phone, I will take it!!!

´ No food or drinks (except for water) are allowed in my room. Finish your food outside before you enter my classroom.

Testing the Difference Between ProportionsUsing the 𝑧 Test

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Remember the steps: 1. Hypothesis

𝐻#: 𝑝& − 𝑝( = 0𝐻&: 𝑝& − 𝑝( ≠ 0

𝐻#: 𝑝& − 𝑝( = 0𝐻&: 𝑝& − 𝑝( < 0

𝐻#: 𝑝& − 𝑝( = 0𝐻&: 𝑝& − 𝑝( > 0

Remember the steps: 2. Critical Values

´When dealing with proportions we use 𝑧-scores.

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Remember the steps: 2. Critical Values

´ If you do not remember or have limited access to the critical values for the 3 most common levels of significances, you can always use your 𝑧-table for any level.

Remember the steps: 3. Test Values

Tip: 20 out 80 students like the food in the cafeteria.

The 𝑋 value is 20.The 𝑛 value is 80.The �̂� value is the proportion (#

1#= 0.25.

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Remember the steps: 3. Test Values

´Let’s practice with the test value. Assume that we just want to see if the proportions are different. Find the test value for the following data:I. 𝑛& = 48, 𝑋& = 34; 𝑛( = 48, 𝑋( = 40II. 𝑛& = 75, 𝑋& = 28; 𝑛( = 75, 𝑋( = 30III. 𝑛& = 100, 𝑋& = 50; 𝑛( = 60, 𝑋( = 35IV. 𝑛& = 144, 𝑋& = 12; 𝑛( = 48, 𝑋( = 8

4. DecisionØ For right-tailed: Reject if the test value is greater than critical

value.Ø For left-tailed: Reject if the test value is less than the critical

value. Ø For two-tailed: Reject if the test-value is greater or less than the

positive and negative critical values, respectively. Ø For p-values: Reject if p-value is less than alpha.

5. ConclusionØ Focus on where the claim is.Ø Enough evidence to reject the claim…Ø Enough evidence to support the claim…

Remember the steps: 4 and 5…

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Learning-by-doing: Together #1

Learning-by-doing: Together #1

Step 1 State the hypotheses and identify the claim.

𝐻#: 𝑝& − 𝑝( = 0 [claim]𝐻&: 𝑝& − 𝑝( ≠ 0

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Learning-by-doing: Together #1

Step 2 Find the critical values.

Since 𝛼 = 0.05, the critical values are ±1.96.

Learning-by-doing: Together #1

Step 3 Compute the test value.

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Learning-by-doing: Together #1

Step 3 Compute the test value.

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Learning-by-doing: Together #1

Step 4 Make the decision.

Reject the null hypothesis, since −2.7 < −1.96.

Learning-by-doing: Together #1

Step 5 Summarize the results.

There is enough evidence to reject the claim that thereis no difference in the proportions of small and large nursing homes with a resident vaccination rate of less than 80%..

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Learning-by-doing: Your turn #2

Learning-by-doing: Your turn #2

Check your numbers!

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𝐻#: 𝑝& − 𝑝( = 0 𝐻&: 𝑝& − 𝑝( ≠ 0  [claim]

Partner/Independent Work

´ Work on your worksheet. ´ Do a couple with a partner and at least one by yourself. ´ Show it to me when you are done.

2/14/18

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DO NOW

´ UNIT 6 exam is on Monday, February 19. ´ Find your number in the lists posted in the bulletin boards

(back wall and main entrance).´ Find the seat with your number and that will be your seat from

now on. ´ SILENCE YOUR PHONE and put it in the pocket that has your

number in the bulletin board (back wall).´ NO EXCEPTIONS and NO PHONES in backpack. If I see your

phone, I will take it!!! ´ No food or drinks (except for water) are allowed in my room.

Finish your food outside before you enter my classroom.

Interpreting OutputsUsing p-values, test values and confidence intervals

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Using P-valuesP-value ≤ 𝜶: The difference between the means is statistically significant (Reject H0).

Ø If the p-value is less than or equal to the significance level, the decision is to reject the null hypothesis.

Ø You can conclude that the difference between the population means is statistically significant.

Ø Interpret it based on where the claim is.

P-value > 𝜶: The difference between the means is not statistically significant (Fail to reject H0)

Ø If the p-value is greater than the significance level, the decision is to fail to reject the null hypothesis.

Ø You do not have enough evidence to conclude that the difference between the population means is statistically significant.

Ø Interpret it based on where the claim is.

Example #1

´ Test the claim that there is no difference in the mean rating between two hospitals.

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Example #2

´ Test the claim that the population mean for Variable 1 is significantly higher than the population mean for Variable 2.

Using test values (when p-value is not given)

Identify the test value: Highlight it or write it down.

Find the critical value: Using 𝑧-table or 𝑡-table.Ø Identify the level of significance (𝛼).Ø Decide whether your are using a using 𝑧-table or a 𝑡-table.Ø Find your critical value.

Make a decision: reject or do not reject the null.Ø For right-tailed: Reject if the test value is greater than critical value.Ø For left-tailed: Reject if the test value is less than the critical value. Ø For two-tailed: Reject if the test-value is greater or less than the

positive and negative critical values, respectively.

Don’t forget your conclusion!: There is (or is not) enough evidence to support (or to reject) the claim that...

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Example #1

´ Test the claim that there is a difference in the price of gasoline between State A and State B.

Example #2

´ Test the claim that those who stayed had a higher science grade point average than those who left. Use a 0.05 level of significance.

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Using Confidence Intervals

• There is a relationship between confidence intervals and hypothesis testing.

• When the null hypothesis is rejected in a hypothesis-testing situation, the confidence interval for the mean using the same level of significance will not contain the hypothesized mean.

• Likewise, when the null hypothesis is not rejected, the confidence interval computed using the same level of significance will contain the hypothesized mean.

Example #1

´ Test the claim that over time FC Barcelona is one average point in standings better than Real Madrid.

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Example #2

´ Test for a difference in the student vaccination rates between middle school and high school.

Example #3

´ Test the claim that the proportions are different in each of the following cases.

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Partner/Independent Work

´Work on your worksheet. This should be completed in 10-15 minutes.

´Work on your cheat sheet for Unit 6 Exam.

´Show it to me when you are done.