STAT homework by minitab

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STAT 108 - Lab 8

Lab 8 due at the beginning of lab on Tuesday, April 19

Objectives  Assess conditions for, conduct, interpret, and report results of t-tests in Minitab

Correct presentation of steps in conducting and reporting results of t-tests

When conducting and reporting the results of t-tests, include the following steps. This applies to the lab Examples and

to the problems in the lab assignment. Detailed examples of how to do the reporting are given in Examples 1 and 2.

1. Verify that conditions for the t-test have been met:

a. Samples selected randomly or from a randomized experiment (you will need to be given some

information about the sampling or experimental procedures)

b. Each observation should be considered independent

c. The sampling distribution considered normally distributed. Either the population 𝑋 is normal (assessed

with normal probability plots or given in description, described in Example 3) OR sample size at least 30.

2. Identify the response variable and state the null and alternative hypotheses.

3. State the level of significance (will be provided).

4. Perform the t-test in Minitab and report: the test statistic, the degrees of freedom, and the p-value (usually this

just amounts to copy/pasting the Minitab output).

5. State the conclusion of the test along with the p-value in two ways: 1-either reject or fail to reject the null

hypothesis, 2-state the conclusion written within the context of the problem.

Definition and interpretation of hypothesis testing error types

The hypothesis testing decision table is given below, for reference.

Decision based on sample (known to us)

Reject the null hypothesis Fail to reject null hypothesis

TRUTH (unknown to us)

Null hypothesis is True Type I error (Probability = 𝛼)

Correct decision

Null hypothesis is false (H1 is true)

Correct decision Type II error

 𝛼: Probability that sample provides evidence (known) to reject Ho, given that the null hypothesis is true

(unknown). This is the fraction of random samples that lead us to incorrectly reject the null hypothesis (test

statistic is large by random chance, not large because the null hypothesis is false).

 p-value: Probability that the sample is consistent with the null hypothesis, in other words, the probability that

due to sampling error alone (chance), the samples could give us a test statistic at least as extreme as the one

observed, if the null hypothesis is true.

We reject the null hypothesis when p-value < 𝛼. We never know if we made a correct or incorrect conclusion when

working with sample data.

This reiterates the

direction of the H1 and

the value of 𝜇0 you

provided to Minitab

The p-value is

displayed here.

You will base

your conclusion

on this value

This is the test-

statistic 𝑡0

Lab Examples From our Moodle site, download the Minitab file called “Lab_8_t_test_data.MPJ”.

Example 1. t-test; left-sided alternative The column “Daily Calories” contains the daily caloric intake of 500 randomly chosen adults from a rural community in a

developing country. We wish to know if we can conclude that the mean daily calorie intake in this rural community is

less than 2000. We considered each adult as an independent observational unit. Use 𝛼 = 0.05.

Before you conduct the t-test, draw a normal probability plot of the sample to

check assumptions related to normality and/or sample size. Click: Graph>

Probability Plot> Single then enter Daily Calories into the Graph Variables box,

OK. In this case, the sample shows mild departures from normality (describe

them in your lab), but we do not need to be concerned with those since the

sample size is very large.

To conduct the t-test for the mean in Minitab, click Stat>Basic Statistics>1 Sample t, select One or more samples, each

in a column, and enter the column Daily Calories in the second box. Check the box for Perform hypothesis test, and

enter the hypothesized mean of 2000. The alternative hypotheses and significance level can be set under Options. For

this problem, use a left-sided alternative hypothesis (“Mean < hypothesized mean”) and α = 0.05 (confidence = 95%;

default). OK>OK.

To present this Example, I will follow the steps on Page 1 for performing the test and reporting the results. Here is a

complete example of how to do this (you would do this in writing in your lab report).

1. We were told that the data were obtained by random sampling and are independent. The distribution of the

calorie intake is not normal, but the sample size is very large (n = 500). Thus the conditions for the t-test have

been met.

2. The response variable is the calorie intake for adults in the rural community in the developing nation.

Ho: μ = 2000

H1: μ < 2000

3. 𝛼 = 0.05

4. You can just copy/paste the Minitab output:

One-Sample T: Daily Calories

Test of μ = 2000 vs < 2000

Variable N Mean StDev SE Mean 95% Upper Bound T P

Daily Calories 500 1978.43 191.22 8.55 1992.52 -2.52 0.006

5. We reject the null hypothesis since the p-value = 0.006. There is sufficient evidence to conclude that the mean daily caloric intake of adults in the rural community in the developing country is less than 2000.

Note: You do NOT need to calculate the Test Statistic by hand, Minitab does this for you and it is provided in the output.

Just use the p-value to make your conclusions. Remember… “When the p-value is low the null must go!”

Example 2. t-test; two-tailed alternative The column Heart Rate 10 mins treadmill contains the recovering heart rate in beats per minute (bpm) of 25 randomly

chosen college students after spending 10 minutes on a treadmill. Let’s test whether the recovering heart rate is

significantly different from 100 bpm at 𝛼 = 0.10.

Before you conduct the t-test, draw a normal probability plot of the sample to

check assumptions related to normality and/or sample size. In this case, the

sample size is not particularly large (n = 25), but the data appear to be normally

distributed.

To conduct the t-test for the mean in Minitab, click Stat>Basic Statistics>1

Sample t, select One or more samples, each in a column, and enter the column Heart Rate in the second box. Check the

box for Perform hypothesis test, and enter the hypothesized mean of 100. The alternative hypotheses and significance

level can be set under Options. For this problem, use a two-sided alternative hypothesis (“Mean ≠ hypothesized mean”)

and α = 0.10 (confidence = 90%). OK>OK.

To present this Example, I will follow the steps on Page 1 for performing the test and reporting the results. Here is a

complete example of how to do this (you would do this in writing in your lab report).

1. We were told that the data were obtained by random sampling and we can fairly assume that the individual

heart rates are independent. The distribution of the heart rates appears to be normally distributed. Thus the

conditions for the t-test have been met.

2. The response variable is the recovering heart rate of college students after spending 10 minutes on a treadmill.

Ho: μ = 100

H1: μ ≠ 100

3. 𝛼 = 0.10

4. You can just copy/paste the Minitab output:

One-Sample T: Heart Rate 10 mins treadmill

Test of μ = 100 vs ≠ 100

Variable N Mean StDev SE Mean 90% CI T P

Heart Rate 10 mins tread 25 98.76 12.96 2.59 (94.33, 103.19) -0.48 0.637

5. We fail to reject the null hypothesis since the p-value = 0.637. There is insufficient evidence to conclude that the mean recovering heart rate is different than 100 bpm for college students after 10 minutes on a treadmill.

Example 3. Assessing t-test conditions: the test for normality The column “Daily Calories” contains the daily caloric intake of 500 randomly chosen adults from a rural community in a

developing country. Before conducting any t-test, we need to draw a normal probability plot of the samples to check

conditions related to normality and/or sample size.

To do this, select Graph>Probability Plot>Single. Enter Daily Calories in the box, and then select OK.

Notice the p-value in the summary box inside the plot.

That p-value is for what is called the “test

for normality”. Its null hypothesis is that

the sample came from a normal

distribution. The p-value is 0.021, which

means that we would reject the

assumption of normality at α = 0.05.

However, we can conclude that the conditions of the t-test have been met for the “Daily Calories” sample (because of

the very large sample size and only mild skewness).

We noticed that the normality test would fail to reject the null hypothesis of normality in the “Heart Rate 10 mins

treadmill” sample. This was critical since the sample size was n = 25, which is less than 30.

Problems to submit for Lab 8:

Please provide thorough answers to each question below in a document (Work, Google docs, etc). Please

USE COMPLETE SENTENCES AND APPROPRIATE TERMINOLOGY and notation in your responses where

applicable. You may work together on these assignments, but each person’s work should be their own.

Notes regarding this lab assignment:

This Lab assignment is larger and more detailed than past Lab assignments. This is because it is probably the single most

important assignment of the semester in terms of content and presentation. Read the problems carefully, do all parts of

the problem, and refer to the examples above to write your answers in the proper form. All of the information and

methodology you need to work the problems is contained in one or more of the examples above, so study them

carefully to obtain some guidance.

This reiterates the

direction of the alternative

hypothesis and the value

of 𝜇0 you provided to

Minitab

The p-value is displayed

here. You will base your

decision on this value

This is the test-statistic 𝑡0.

Calculated as

𝑡0 = 208.4 − 198

9.38/√10

1. Michael Sullivan, son of the author of our textbook, decided to enroll in a reading course that allegedly increases

reading speed and comprehension. Prior to enrolling in the class, Michael read 198 words per minute (wpm).

The data in the column reading speed (wpm) represents the words per minute read for 10 randomly selected

passages after the course.

a) Draw a normal probability plot of the sample and include this figure in your lab assignment. Explain why the conditions for the use of the t-test are met. (Remember to include a statement about sampling and independence).

b) State the response variable, null hypothesis, and alternative hypothesis, c) Suppose we want the Type I error rate to be 10%. State the level of significance for the test d) Perform a hypothesis test to address whether the class is effective at increasing reading speed and

comprehension the 𝛼 = 0.10 level of significance? Basically, just copy-paste the Minitab output e) State your conclusion to the t-test in two ways:

o A statistical conclusion (reject the null hypothesis or do not reject the null hypothesis) and report the p- value.

o A conclusion within the context of the problem (using complete sentences as illustrated in the Examples above)

f) If you did everything correctly, you should have rejected the null hypothesis. Do you know whether you

made an error in reaching this conclusion? If you did make an error, what Type was it?

Below is the Minitab output you should encounter for Problem 1 part d.

One-Sample T: Reading Speed (wpm)

Test of μ = 198 vs > 198

Variable N Mean StDev SE Mean 90% Lower Bound T P

Reading Speed (wpm) 10 208.40 9.38 2.97 204.30 3.50 0.003

2. An engineer wants to determine whether a pH meter is biased. She uses the meter to measure the pH in 14

randomly chosen neutral substances (pH = 7.0) and obtains the data contained in the column pH level.

b) State the response variable, null hypothesis, and alternative hypothesis. c) Suppose we want the Type I error rate to be 5%. State the level of significance for the test. d) Perform a hypothesis test to address whether the pH meter is biased at the 𝛼 = 0.05 level of significance?

Basically, just copy-paste the Minitab output e) State your conclusion to the t-test in two ways:

o A statistical conclusion (reject the null hypothesis or do not reject the null hypothesis) and report the p- value.

o A conclusion within the context of the problem (using complete sentences as illustrated in the above Examples)

f) Do you know whether you made an error in reaching this conclusion? If you did make an error, what Type

was it?

3. An important factor in judging the value of a publicly traded company is its earnings. Therefore, stock analysts

will attempt to predict the earning for a company. The following data represent the deviation from the actual

earnings for General Electric for 2007. Negative values mean the actual earnings were less than predicted, while

positive values mean that the actual earnings were more than predicted. So the data are calculated as:

Deviation = Actual earnings – Predicted earnings

A normal probability plot indicates that the data could have come from a population that is normally distributed.

The sampled values can also be regarded as independent. A researcher wanted to determine if, on average, the

stock analyst’s earnings estimates were on target. The following display shows output from Minitab.

One-Sample T: GE Deviation from earnings

Test of μ = 0 vs ≠ 0

Variable N Mean StDev SE Mean 95% CI T P

Earning deviation 12 -0.01500 0.02316 0.00669 (-0.02971, -0.00029) -2.24 0.046

a) Explain why the researcher tested H0: µ = 0 versus H1: µ ≠ 0? Discuss the two-tailed alternative hypothesis and the value of zero.

b) Based on the output, what should the researcher conclude if she uses an 𝛼 = 0.05 level of significance? State the two conclusions.

c) The researcher’s boss does not think it would be good for the industry to state that the analyst’s earnings estimates are incorrect, on average. So he tells the researcher to change her level of significance to 𝛼 = 0.01. Explain why this is an example of bad statistics.

4. Describe, in detail, how the contents of this Lab Assignment and the Lab Examples connect with the material we

have covered during recent lecture classes.

 To receive full credit, write a few/several sentences using appropriate statistical terminology and utilize proper grammar and spelling.