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Hypothesis Testing on Population Means: Activity 4, Exams of Mathematics

Saint Mary's CollegeMathematics

An activity for students to learn about hypothesis testing on population means. The activity involves working in teams to complete exercises and assessing team performance. Examples of hypothesis tests for mean yields of corn and accidents at an intersection, and includes instructions for calculating test statistics and determining critical values or p-values. Students are expected to use their textbook and calculator for the exercises.

Typology: Exams

Pre 2010

Uploaded on 08/05/2009

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Statistical Applications ACTIVITY 4: Hyp othesis Tests on the Mean
Why
The predominant (but far from the only) mode of decision-making in statistics is theā€œtest of
significanceā€. The mechanics vary between situations, but the underlying logic is the same. we
need to establish the basic ideas ā€“ using tests on a population mean.
LEARNING OBJECTIVES
1. Work as a team, using the team roles
2. Understand the logical basis of tests of significance - focusing on tests on the mean.
3. Be able to use test procedures to make a statistical decision.
CRITERIA
1. Success in completing the exercises.
2. Success in answering questions about the model
3. Success in working as a team
RESOURCES
1. The ā€œHYPOTHESIS TESTING USING Ī±AND Ī²(chapters 9,10 and 11)ā€™ handout from Monday
2. for mechanics: the document ā€œHypothesis testing - generalitiesā€ attached to this activity sheet
3. Your Text - especially sections 9.1 through 9.4
4. The model below
5. Your Calculator
6. 40 minutes
PLAN
1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 (5
minutes)
2. Work through the exercises given here - be sure everyone understands all results (30 minutes)
3. Assess the teamā€™s work and roles performances and prepare the Reflectorā€™s and Recorderā€™s reports
including team grade (5 minutes).
4. Be prepared to discuss your results.
MODELS
1. A new fertilizer is being tested for use on corn; the tests will be carried out in an area in which
the mean yield has been 142 bushels per acre, . We will assume that corn yields are known to be
approximately normally distributed. Naturally, we will want to know if the new fertilizer increases
the yields. A sample of 5 plots with the new fertilizer gives yields of 147.5, 140.6, 155.3 , 156.2, and
148.4 bushels per acre. Does this show that the new fertilizer produces an increase in mean yield?
[Test at 5% level]
I Population - corn fields fertilized with the new fertilizer. Variable X = corn yield (bu/A). Test
is on mean. Test is :
H0:Āµ= 142
Ha:Āµ > 142
II Test statistic is t=ĀÆxāˆ’142
sx/āˆšnwith df = nāˆ’1
III Critical value approach: Reject H0if sample t>t.05 = 2.132 (with df = 4)[table p. 920]
IV t=149.6āˆ’142
6.38/āˆš5= 2.66
V Reject H0and support Ha
1
pf3

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Download Hypothesis Testing on Population Means: Activity 4 and more Exams Mathematics in PDF only on Docsity!

Statistical Applications ACTIVITY 4: Hypothesis Tests on the Mean

Why

The predominant (but far from the only) mode of decision-making in statistics is theā€œtest of significanceā€. The mechanics vary between situations, but the underlying logic is the same. we need to establish the basic ideas ā€“ using tests on a population mean.

LEARNING OBJECTIVES

  1. Work as a team, using the team roles
  2. Understand the logical basis of tests of significance - focusing on tests on the mean.
  3. Be able to use test procedures to make a statistical decision.

CRITERIA

  1. Success in completing the exercises.
  2. Success in answering questions about the model
  3. Success in working as a team

RESOURCES

  1. The ā€œHYPOTHESIS TESTING USING Ī± AND Ī² (chapters 9,10 and 11)ā€™ handout from Monday
  2. for mechanics: the document ā€œHypothesis testing - generalitiesā€ attached to this activity sheet
  3. Your Text - especially sections 9.1 through 9.
  4. The model below
  5. Your Calculator
  6. 40 minutes

PLAN

  1. Select roles, if you have not already done so, and decide how you will carry out steps 2 and 3 ( minutes)
  2. Work through the exercises given here - be sure everyone understands all results (30 minutes)
  3. Assess the teamā€™s work and roles performances and prepare the Reflectorā€™s and Recorderā€™s reports including team grade (5 minutes).
  4. Be prepared to discuss your results.

MODELS

  1. A new fertilizer is being tested for use on corn; the tests will be carried out in an area in which the mean yield has been 142 bushels per acre,. We will assume that corn yields are known to be approximately normally distributed. Naturally, we will want to know if the new fertilizer increases the yields. A sample of 5 plots with the new fertilizer gives yields of 147.5, 140.6, 155.3 , 156.2, and 148.4 bushels per acre. Does this show that the new fertilizer produces an increase in mean yield? [Test at 5% level]

I Population - corn fields fertilized with the new fertilizer. Variable X = corn yield (bu/A). Test is on mean. Test is : H 0 : Ī¼ = 142 Ha : Ī¼ > 142 II Test statistic is t = (^) sĀÆxxāˆ’/^142 āˆšn with df = n āˆ’ 1 III Critical value approach: Reject H 0 if sample t > t. 05 = 2. 132 (with df = 4)[table p. 920] IV t = (^1496). 38.^6 /āˆ’āˆš^1425 = 2. 66 V Reject H 0 and support Ha

VI The sample does give evidence at the .05 level that the mean yield is increased by the fertilizer.

Using the p-value approach, weā€™d have:

III t = (^1496). 38.^6 /āˆ’āˆš^1425 = 2. 66 df = 4

IV p = P (t > 2 .66), approximating from the table [p. 920] p is between .05 and .025 (since t is between t.05 = 2. 132 and t.025 = 2. 776 ) [Calculator/ computer gives p =. 028 ]

Steps V, VI are the same.

  1. The number of accidents at a certain hazardous intersection has varied, with a mean 2.2 accidents per week. Because of construction work on different roads and the opening of other roads, the highway department thinks the rate may have changed - but donā€™t know which way. A sampling over twenty weeks gives a mean 2.12 accidents and standard deviation 1.4 accidents (per week). [We will assume the number of accidents per week is symmetric enough that the sample means will be approximately normally distributed] Does this provide evidence that the mean number of accidents has changed?

I Population: weeks [note our sample is of twenty weeks]. Variable: X = number of accidents at this intersection (with the new conditions) Parameter: Mean. Test is: H 0 : Ī¼ = 2. 2 Ha : Ī¼ 6 = 2. 2 II Statistic t = (^) sĀÆxxāˆ’/^2 āˆš.^2 n df = n āˆ’ 1 III To test at .05 level: Reject H 0 if sample t > t. 025 or if sample t < āˆ’t. 025 -from table, with df = 19, t. 025 = 2. 093 IV sample t = 21 ..^124 /āˆ’āˆš^220.^2 = āˆ’. 26 V We do not reject H 0 the sample does not provide evidence at the .05 level that the mean number of accidents has changed.

Calculating the p-value, from the tabel we can only tell that t does not even reach t. 20 , so we know that p > 2(.20) =. 40. Calculator gives p =. 80. Both methods are saying that this amount of change is easily explained as usual variability - no evidence of a change in the basic situation.

EXERCISE Show all the setup - identify population, variables,etc. Write your conclusion in words.

  1. The faculty at Ino U. claim that students spend an average of 3 hours per day studying. The Student Academic Council believes this figure is too low, and undertakes a study intending to show that the average study time for all students is more than 3 hours per day. The Academic Council selects a simple random sample of 30 students and obtains a mean study time, for the sample, of 3.5 hours per day [Ino U. is not a very competitive college], standard deviation 1.5 hours Now they need to analyze the data.
  2. A soft-drink bottler sells ā€œone-literā€ bottles of soda. The state Consumer Products Division is concerned that the bottler may be short changing the customers. Thus the CPD will test to see if the mean amount in a bottle is less than 1000 ml. Carry out the test. The contents of the 30 bottles of soda selected by the CPD are as follows: (ml): 1025 977 1018 975 977 990 959 957 1031 964 986 914 1010 988 987 1028 989 1001 984 974 1017 1060 1030 991 999 997 996 1014 946 995
  3. The general partner of a limited partnership firm has told a potential investor that the mean monthly rent for three-bedroom homes in the area is $525. The investor wants to check out this claim on her own (see whether the real mean is different or as claimed) She obtains the monthly rental charges for a random sample of 35 three-bedroom homes, obtaining a mean monthly rent of $581.65 and a standard deviation of $118.73. What conclusion about monthly rents will she reach?

READING ASSIGNMENT (in preparation for next class) Read Chapter 9 (9.6ā€“9.7) - Decision-making and types of errors

SKILL EXERCISES:Use your calculator or Minitab for number -crunching [Minitab will carry out hypoth- esis tests when you have actual data to work with] but you have to write the hypotheses and conclusion. p. 358 #20, p.363 #28, 29, p.368(proportions)#40, 41