Calculus and Above

Calculus Questions? Ask a Mathematician for Answers ASAP

Connect one-on-one with {0} who will answer your question

Explain these concepts: 1. Degree of freedom 2. Critical…

Customer Question

Explain these concepts: 1. Degree of...

Explain these concepts:

1. Degree of freedom
2. Critical values for Z and t test
3. Z - test and t-test

Submitted: 8 years ago.Category: Calculus and Above

Show Less

Ask Your Own Calculus and Above Question

Answered in 49 minutes by:

3/8/2012

Tutor: Ryan, Engineer replied 8 years ago

Ryan, Engineer

Category: Calculus and Above

Satisfied Customers: 9,218

Experience: B.S. in Civil Engineering

Verified

Hi,

Thank you for using JustAnswer.

1. Degree of freedom:

A degree of freedom is a comparison between the data that is independent of the other comparisons. For instance, when calculating the standard deviation, you calculate the residual values (x(i) - x-bar). For a set of n values, there will be n residuals. There are only n - 1 independent residuals though, due to the constraint that the sum of all of the residuals is equal to zero.

Another way to look at this is that you could independently select n - 1 values. Once that is done, the last value is already determined by the constraint. Since you only have n - 1 independent choices, there are n - 1 degrees of freedom.

2. Critical values for Z and t test:

The critical values in a Z or t test are the values which define the rejection region under the normal distribution or Student's t-distribution, respectively. In a one-tailed test, the area under the curve to the right of a positive critical value, or to the left of a negative critical value, is equal to the level of significance of the test. In a two-tailed test, the sum of the area in both tails (to the right of the positive critical value and to the left of the negative critical value) is equal to the level of significance of the test.

3. Z-test and t-test:

The Z-test and t-test are statistical tests which compare an observed value against a hypothesized value and determine if the difference is significant, depending on the chosen level of significance. The Z-test uses the normal distribution, and is appropriate for use when either the standard deviation of the population is known, or when the sample size is 30 or higher. The t-test uses the Student's t-distribution, and is appropriate for use when the conditions required for the Z-test are not met.

In each case, the difference between the observed value and the hypothesized value is divided by the standard error, which is equal to the standard deviation divided by the square root of the size of the sample. (For an individual observed value, the sample size is 1, so the standard deviation divided by the square root of 1 is just the standard deviation itself.) The resulting value is then compared with the critical value mentioned above, and a decision is made as to whether to reject or fail to reject the null hypothesis.

It is important to grasp the concept that the null hypothesis is never "accepted". It is either rejected or not rejected. When it is not rejected, there is simply not enough evidence to conclude convincingly that the observed value is significantly different from the hypothesized value.

Please feel free to ask if you need additional information, or if you have any additional questions about these topics.

Thanks,

Ryan

Customer reply replied 8 years ago

Can you help me with these questions, please?
Thank you.
1. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation of 30 items. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day.

The statistisians should answer the question: Is the difference of 230 and 200 statistically significant?

What is the correct formula of the Null Hypothesis for this study?

2.. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation of 30 items. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day.

What would be the sample error in this study?

3. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation of 30 items. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day.

What would be the test-statistics in this study?

4. Identify the correct formula for Z-test, if the mean of the population is μ, the mean of the sample is x̅

the standard deviation of the population is σ, and the size of the sample is n.

5.

The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation of 30 items. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day.

The statistisians should answer the question: Is the difference of 230 and 200 statistically significant?

If the Z-test is 10, and Z-critical = + or - 1.96, what is the result of the test.

b. Z-critical < Z-test, therefore, I will reject the null. The difference of productivity after the training is statistically significant.

6. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation unknown. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day, with standard deviation 100. The statistisians should answer the question: Is the difference of 230 and 200 statistically significant? Calculate the sampling error?

7. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation unknown. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day, with standard deviation 100. The statistisians should answer the question: Is the difference of 230 and 200 statistically significant?

What is the degree of freedom in this case?

8. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation unknown. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day, with standard deviation 100. The statistisians should answer the question: Is the difference of 230 and 200 statistically significant?

If the level of significance is alpha = 0.05, what would be the critical value for the t-test?

9. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation unknown. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day, with standard deviation 100. The statistisians should answer the question: Is the difference of 230 and 200 statistically significant?

Calculate the t-test?

10.

Can you help me with these questions, please?
Thank you.

1. The researcher is testing a hypothesis about the productivity. In the last three years, the average productivity of workers was 200 items per day, with standard deviation of 30 items. To increase the productivity, managers conducted special training. After the training, a random sample with size 100 was selected from the population of all workers. The average productivity for this sample was 230 items per day.

The statistisians should answer the question: Is the difference of 230 and 200 statistically significant?

What is the correct formula of the Null Hypothesis for this study?

What would be the sample error in this study?

What would be the test-statistics in this study?

4. Identify the correct formula for Z-test, if the mean of the population is μ, the mean of the sample is x̅

the standard deviation of the population is σ, and the size of the sample is n.

5.

The statistisians should answer the question: Is the difference of 230 and 200 statistically significant?

If the Z-test is 10, and Z-critical = + or - 1.96, what is the result of the test.

b. Z-critical < Z-test, therefore, I will reject the null. The difference of productivity after the training is statistically significant.

What is the degree of freedom in this case?

If the level of significance is alpha = 0.05, what would be the critical value for the t-test?

Calculate the t-test?

Tutor: Ryan, Engineer replied 8 years ago

Hi,

Can you please check the questions for #6 through #10? They seem to include the same basic data as question #1 through #5. The problem with this is that the t-test is not really appropriate for that scenario, so I'm wondering if it is possible that problems 6 through 10 are supposed to have different information.

Thanks,

Ryan

Customer reply replied 8 years ago

The questions are the same for Q #6 to 10. I believe they are asking for is in the last sentence?

Tutor: Ryan, Engineer replied 8 years ago

Hi,

Ok, here are the solutions:

1. The null hypothesis is H0: µ = 200

2. The sample error is 30

3. The z-statistic; because the standard deviation of the population is known. (And even if it wasn't, the sample size is greater than 30, so the z-statistic would still be used.)

4. z = (x̅ - µ) / (σ / √n)

5. b

6. The sampling error is 30.

7. The number of degrees of freedom is 99

8. At alpha = 0.05, for a two-tailed test with 99 degrees of freedom, the critical t-values are 1.984 and -1.984

9. The test t-value is (230 - 200) / (100 / √100) = 3

Thanks,

Ryan

Customer reply replied 8 years ago

Can you please verify Question 3's answer? Thank you.

Tutor: Ryan, Engineer replied 8 years ago

Hi,

I interpreted #3 to be asking about which type of statistic you would be using for this test, as in choosing between the z-statistic and the t-statistic.

Since the standard deviation of the population is given as 30, it is a known quantity. The z-statistic can be used whenever the standard deviation of the population is known.

If this question is looking for the particular value of the test statistic, then it is 10, as given in #5. The calculation for it is (230 - 200) / (30 / √100) = 10.

Thanks,

Ryan

Ryan, Engineer

Category: Calculus and Above

Satisfied Customers: 9,218

Experience: B.S. in Civil Engineering

Verified

Ryan and 87 other Calculus and Above Specialists are ready to help you

Ask your own question now

Was this answer helpful?

Yes

How JustAnswer works

Describe your issueThe assistant will guide you

Chat 1:1 with a tutorLicensed Experts are available 24/7

100% satisfaction guaranteeGet all the answers you need

Related Calculus and Above Questions

Confidence level 99%; n = 17; σ is unknown; population

Confidence level 99%; n = 17; σ is unknown; population appears to be normally distributed.… read more

Keith Pelletier

Advanced Math Expert

Master's Degree

275 satisfied customers

Part D and E of the same question 1. Levine et al. (2005)

part D and E of the same question 1. Levine et al. (2005) found that people are especially likely to help others who they regard as part of their own group. Suppose a researcher attempted to replicate… read more

Scott

MIT Graduate

Master's Degree

17,852 satisfied customers

Solve the problem. For large numbers of degrees of freedom,

Solve the problem. For large numbers of degrees of freedom, the critical values can be approximated using the formula , where k is the number of degrees of freedom and z is the critical value. To fi… read more

Ashleigh T

Educator

Master's Degree

121 satisfied customers

A shipment of small lab mice arrives at an animal care

A shipment of small lab mice arrives at an animal care facility with a nominal weight of 10 g per mouse. The technician wishes to test the hypothesis that the average weight of the mice matches the no… read more

Sandhya

Master's Degree

6,028 satisfied customers

How does this work? and how much does this cost. Statistics

how does this work? and how much does this cost JA: The Advanced Math Tutor can help you get an A on your homework or ace your next test. Tell me more about what you need help with so we can help you … read more

Ryan

Engineer

Bachelor's Degree

9,218 satisfied customers

In an independent samples t-test with groups of 28 and 25,

in an independent samples t-test with groups of 28 and 25, how many degrees of freedom are there… read more

Ryan

Engineer

Bachelor's Degree

9,218 satisfied customers

When performing a rank correlation test, one alternative to

When performing a rank correlation test, one alternative to using the Critical Values of Spearman's Rank Correlation Coefficient table to find critical values is to compute them using this approximati… read more

GLENN GILONGOS

Master of Mathematics

637 satisfied customers

I need help with statistic problems. There are about 25 questions.

I need help with statistic problems. There are about 25 questions. I have answered some but keep getting the anwers wrong. If you can please help. Let me know what the cost will be. Thanks… read more

Math-John

30 satisfied customers

You decide to conduct a survey of a sample of 25 members of

You decide to conduct a survey of a sample of 25 members of this year's graduating class and find that the average GPA is 3.2. The standard deviation of the sample is 0.4. Over the last ten years, the… read more

Ryan

Engineer

Bachelor's Degree

9,218 satisfied customers

Conduct a two-tailed hypothesis test given the information

Conduct a two-tailed hypothesis test given the information below. Assuming that the population variances are equal for male and female Grade Point Averages (GPAs), use the following sample data to tes… read more

Ryan

Engineer

Bachelor's Degree

9,218 satisfied customers

A researcher believes that certain types of doctors are happier

A researcher believes that certain types of doctors are happier than other doctors. One theory is that certification level is an important factor their overall sense of happiness. To investigate this,… read more

Ryan

Engineer

Bachelor's Degree

9,218 satisfied customers

For Ryan, Hello Ryan I was hoping you could assist me with

For Ryan, Hello Ryan I was hoping you could assist me with several math probelms that I have to do. Do you have time to assist me?… read more

Ryan

Engineer

Bachelor's Degree

9,218 satisfied customers

E = 1.993943 9/√72=2.114896. Can you breakdown how you arrived

E = 1.993943 9/√72=2.114896. Can you breakdown how you arrived at this answer? Did you multiply the 1.993943 by the answer of 9 divided by the sq. rt of 72?… read more

Math-John

30 satisfied customers

1.A coin operated drink machine was designed to discharge a

1.A coin operated drink machine was designed to discharge a mean of 8 ounces of coffee per cup. Suppose that we want to carry out a hypothesis test to see if the true mean discharge differs from 8. St… read more

Chirag

Master's Degree

10,329 satisfied customers

The historical reports from two major networks showed that

The historical reports from two major networks showed that the mean number of commercials aired during prime time was equal for both networks last year. In order to find out whether they still air the… read more

Chirag

Master's Degree

10,329 satisfied customers

A recent study at a local college claimed that the proportion,

A recent study at a local college claimed that the proportion, p, of students who commute more than fifteen miles to school is no more than 15%. if a random sample of 275 students at this college is s… read more

SusanAthena

Master's Degree

102 satisfied customers

1. A regional commuter airline selected a random sample of

1. A regional commuter airline selected a random sample of 25 flights and found that the correlation between the number of passengers and the total weight, in pounds, of luggage stored in the luggage … read more

Ryan

Engineer

Bachelor's Degree

9,218 satisfied customers

find critical value(s) for the indicated t-test, level of significance

find critical value(s) for the indicated t-test, level of significance a, and sample size n. two tailed test, a=0.02, n=6 the critical value(s) are?… read more

Expert

Disclaimer: Information in questions answers, and Show More

Disclaimer: Information in questions, answers, and other posts on this site ("Posts") comes from individual users, not JustAnswer; JustAnswer is not responsible for Posts. Posts are for general information, are not intended to substitute for informed professional advice (medical, legal, veterinary, financial, etc.), or to establish a professional-client relationship. The site and services are provided "as is" with no warranty or representations by JustAnswer regarding the qualifications of Experts. To see what credentials have been verified by a third-party service, please click on the "Verified" symbol in some Experts' profiles. JustAnswer is not intended or designed for EMERGENCY questions which should be directed immediately by telephone or in-person to qualified professionals.

JustAnswer in the News:

What Customers are Saying:

Wonderful service, prompt, efficient, and accurate. Couldn't have asked for more. I cannot thank you enough for your help.
Mary C.Freshfield, Liverpool, UK

This expert is wonderful. They truly know what they are talking about, and they actually care about you. They really helped put my nerves at ease. Thank you so much!!!!
AlexLos Angeles, CA

Thank you for all your help. It is nice to know that this service is here for people like myself, who need answers fast and are not sure who to consult.
GPHesperia, CA

I couldn't be more satisfied! This is the site I will always come to when I need a second opinion.
JustinKernersville, NC

Just let me say that this encounter has been entirely professional and most helpful. I liked that I could ask additional questions and get answered in a very short turn around.
EstherWoodstock, NY

Thank you so much for taking your time and knowledge to support my concerns. Not only did you answer my questions, you even took it a step further with replying with more pertinent information I needed to know.
RobinElkton, Maryland

He answered my question promptly and gave me accurate, detailed information. If all of your experts are half as good, you have a great thing going here.
DianeDallas, TX

< Previous | Next >

Meet the Experts:

rajeevanpillai

Master's Degree

1,335 satisfied customers

B.Sc Mathematics and M.Sc, M Phil Statistics

Mr. Gregory White

Master's Degree

114 satisfied customers

M.A., M.S. Education / Educational Administration

SusanAthena

Master's Degree

102 satisfied customers

Tutor for Algebra, Geometry, Statistics. Explaining math in plain English.

Dr Arthur Rubin

Doctoral Degree

96 satisfied customers

Ph.D. in Mathematics, 1978, from the California Institute of Technology, over 20 published papers

JACUSTOMER-yrynbdjl-

Master's Degree

42 satisfied customers

I have a MS in Mathematics and I have taught Mathematics for 10 years at the college level.

dhouse1940

Master's Degree

33 satisfied customers

BS mathematics, MS biostatistics, 35+ yrs designing & analyzing biological experiments.

Don

Master's Degree

29 satisfied customers

M.S. Astronautical Engineering. Math/Sci/Comp Tutor

< Previous | Next >

Show MoreShow Less