Business Stats (Week 2)

1) A sample of 1,000 visitors to a website finds that these 1,000 visitors spent an average of 36 minutes at the site. A statistical test of the alternative hypothesis that the mean time spent by visitors to this site is less than 40 minutes ends with the statement: We do not have evidence that the mean time spent at the site is less than 40 minutes. Select all statements that are known to be true.

-The P-value for this test is greater than the significance level of the test. -Even though the sample mean is 36, the population mean does not have to be less than 40. -n = 1,000

Margin of Error/Confidence Interval Review

-When you are required to be more precise, you are going to be less confident -When greater error is allowed, you can be more confident

non-significant result

-any observed difference or relationship could have happened by chance -when the p-value is too high for the null hypothesis to be considered false

How to find p-value

-for left-tailed graph: "for shaded region (area to the left of z-score), it is the probability that z< (z-score)" -Example: •Given: null hypothesis: mu > or =.15; alternative hypothesis: mu < .15 •P (z<-1.34)= .0901; p-value is .0901 -for right-tailed graph: "for shaded region (area to the right of z-score), it is the probability that z> (z-score)" -Example: •Given: null hypothesis: mu < or = .43; alternative hypothesis: mu > .43 •P (z > 2.78)—> P (z < -2.78)= .0027; p-value is .0027

p-value definition

-probability of obtaining a sample "more extreme" than the ones observed in your data, assuming that the null hypothesis is true -The probability that, IF the null hypothesis were true, sampling variation would produce an estimate that is further away from the hypothesized value than our data estimate. -tells us how likely it is to get a result if the null hypothesis is true

Which of the following statements are true?

-α is usually set between 0.05 and 0.01 -The probability of committing a Type I error is denoted α. -The probability of committing a Type II error is denoted β.

A sample size of 150 is collected to test the alternative hypothesis that the population means is greater than 75. The average of the 150 observations is 76. Assuming a population standard deviation of 4, what is the P-value for this test?

.0011

Suppose the consequences of a Type I error are very serious. Which of the following values would be the most reasonable for α?

.01

Suppose that we want to test the claim that the mean of a population is greater than 20. Also suppose that the population standard deviation is known to be 7. A sample of 49 measurements reveals a mean of 22. What is the p-value?

.0228

A sample size of 500 is collected to test the alternative hypothesis that the population mean is not 100, the average of the 500 observations is 99.4. Assuming a population standard deviation of 8, what is the P-value for the test?

.0936

A sample size of 32 is collected to test the alternative hypothesis that the population mean is greater than 75. the average of the 32 observations is 75. Assuming a population standard deviation of 4, what is the P-value of this test?

.5

The amount of time spent browsing a company's website (in minutes) is tracked for a random sample of 43 visitors to the site. The average browsing time for these 43 visitors is 4. The company is assuming a population standard deviation of 2. The company's policy on confidence levels requires that confidence intervals be calculated using a z-score of 4. What is the Margin of error for the resulting confidence interval?

.5978

6) A sample size of 77 is collected to test the alternative hypothesis that the population mean is not 100. The average of the 77 observations is 97.2. Assuming a population standard deviation of 8, what is the P-value for this test?

0.0022

5) A sample size of 32 is collected to test the alternative hypothesis that the population mean is greater than 75. The average of the 32 observations is 77. Assuming a population standard deviation of 4, what is the P-value for this test?

0.0023

4) A sample size of 32 is collected to test the alternative hypothesis that the population mean is greater than 75. The average of the 32 observations is 76. Assuming a population standard deviation of 4, what is the P-value for this test?

0.0786

7) A sample size of 500 is collected to test the alternative hypothesis that the population mean is not 100. The average of the 500 observations is 100.3. Assuming a population standard deviation of 8, what is the P-value for this test?

0.4018

The amount of time spent browsing a company's website (in minutes) is to be tracked for a random sample of visitors to the site. The company is assuming a population standard deviation of 2.4. The company's policy on confidence levels requires that confidence intervals be calculated using a z-value of 2.79. The company has decided that the largest margin of error that would still be useful is .2. How many visitors should the company have in their random sample?

1,121

known to be 7. A sample of 49 measurements reveals a mean of 22. What is the value of the test statistic?

2.00

15) The amount of time spent browsing a company's website (in minutes) is to be tracked for a random sample of 91 visitors to the site. The average browsing time for these 91 visitors is 7. The company is assuming a population standard deviation of 4. The company's policy on cinfidence levels requires that the confidence intervals be calulated using a z*-value of 9. What is the MOE for the resulting confidence interval?

3.7738

14) The amount of time spent browsing a company's website (in minutes) is to be tracked for a random sample of visitors to the site. The company is assuming a population standard deviation of 2.8. The company's policy on confidence levels requires that confidence intervals be calculated using a z*-value of 2.51. The company has decided the largest MOE that would still be useful is 0.3. How many visitors should the company have in their random sample?

549.0

13) In general, the calculation of a P-value allows one to answer which of the following questions (select all that apply):

Are the data more than what one would expect by chance alone? Are the data of practical importance?

JMP Step-by-Step:

Go to Help > Sample Data Click on Teaching Scripts > Interactive Teaching Modules Click on Distribution Calculator Find the Square Root of the Sample Size Divide that number (from Step 4) by the Standard Deviation Enter that number (from Step 5) into the Standard Deviation box under "Parameters" Enter the hypothesized mean in the Mean box under "Parameters" Enter the found mean into the "Input Value" box Select the correct Probability Option For "greater than" questions, use "X>q" For "less than" questions, use "X<=q" For "not equal to" questions, if xbar is less than mu, use "x <=q" For "not equal to" questions, if xbar is greater than mu, use "x>q" BONUS: For "not equal to questions, you must double (x2) the given value at the end for a two-sided probability.

Specify the competing hypotheses that would be used in order to determine whether the population proportion is greater than 0.35.

H0: p ≤ 0.35 versus Ha: p > 0.35

Specify the competing hypotheses that would be used in order to determine whether the population mean differs from 15.

H0: μ = 15 versus Ha: μ ≠ 15

An auditor for a small business wants to test the assumption that the mean value of all accounts receivable differs from $550. What are the null and alternative hypotheses for this test?

H0: μ = 550 versus Ha: μ ≠ 550

Suppose an office manager hopes to establish that a new electronic reporting system decreases the time employees spend filling out travel forms. If they spent an average of 20 minutes filling out travel forms in the past, what hypotheses should the manager test?

H0: μ ≥ 20; Ha: μ < 20

An auditor for a small business wants to determine whether the mean value of all accounts receivable is less than $550. What are the null and alternative hypotheses for this test?

H0: μ ≥ 550 versus Ha: μ < 550

How to do problems 4 & 5: take square root of first number, divide standard deviation by that number, take that number as the SD in JMP; Mean is 2nd number in problem, type last number into value bar at bottom right—>

Help, Sample Data, Teaching Scripts, Interactive Teaching Modules, Distribution Calculator

a (alpha) denotes the probability of Type ____ error happening

I

B (beta) denotes the probability of Type ___ error happening

II

A customer support call center collects data in an effort to demonstrate that the center handles more than 2000 calls per hour, on average. What is the call center's alternative hypothesis?

The average number of calls handled per hour is more than 2000.

16) A customer support call center has handled an average of 2,000 calls per hour in the past. The company believes that changes in the company's products and websites may have changed the number of calls handled by the call center, but they do not know if the average number of calls has gon up or down. They collect data in an effort to demonstrate that the center no longer handles 2,000 calls per hour, on average. What is the call center's alternative hypothesis?

The average number of calls handled per hour is not 2,000.

11) A manufacturing company would like to demonstrate that the company produces less than 4 defective units per 10,000 units manufactured, on average. What is the firm's alternative hypothesis?

The average number of defective units per 10,000 is less than 4.

9) Balloonduck.com is collecting data to test the alternative hypothesis that the average time needed for a poster to obtain a response to a posted question is less than 10 minutes. What is the null hypothesis?

The average response time is greater than or equal to 10 minutes.

12) Some people at your company believe salaries are lower, on average, than the national average for similar companies while others believe salaries are higher, on average, than the national average salary. Eveyone agrees to collect data to test the alternative hypothesis that the average salary at your company is not equal to the average nationwide salary for similar companies. What is the null hypothesis?

The average salary at your company is equal to the average nationwide salary for similar companies.

10) Every month a shipping company collects data to estimate the mean time required to process an order. The company would like to use the data to prove that the mean processing time is greater than 30 minutes. What is the company's alternative hypothesis?

The mean processing time is greater than 30 minutes.

An investment firm would like to collect data to demonstrate to potential investors that the firm's mean rate of return is greater than 8%. What is the firm's alternative hypothesis?

The mean rate of return is greater than 8%

True or false: For a given sample size n, the chances of a Type I error can only be reduced at the expense of a higher chance of Type II error.

True

While performing a statistical analysis, we calculate a P-value test that is less than our pre-specified significance level. What conclusion should we reach?

We should accept the null hypothesis

Suppose we are testing H0: μ = 10 versus Ha: μ ≠ 10. When does a Type I error occur?

When we decide μ ≠ 10 when it is actually = 10

In problems 4 & 5, the mean is

X in JMP

Suppose you are bitten by a strange dog and don't know if the dog has rabies. Your null hypothesis is that the dog has rabies and therefore you should get a rabies shot; your alternative hypothesis is that the dog does not have rabies and therefore you should skip the shot. What happens to you if you make a Type II error?

You get a painful shot you don't really need.

The _______________ hypothesis, denoted Ha, is the statement that will be accepted only if there is convincing sample evidence that it is true.

alternative or research

Suppose a company has developed a new glass to be used in cell phone screens. If the company claims the new glass can withstand more punishment than the glass in current use, then this claim should be made the _________ hypothesis and the statement that the glass really is no better than the current glass should be made the ________ hypothesis.

alternative, null

2) A company performs a statistical test for the alternative hypothesis that the mean time spent by visitors at the company's website is more than 30 minutes. After calculating the P-value, the company fails to reject the null hypothesis. The interpretation of this conclusion is "We _____ evidence that the mean time spent by visitors at the company's website is ______ 30 minutes."

do not have, more than

for subsequent two problems, where it says the alternative hypothesis does NOT equal a value,

do the same thing as 4 & 5, except multiply final answer by 2

all values in shaded area (on the side away from the null hypothesis) are more what?

extreme

8) A P-value near 1 proves that the null hypothesis is true.

false

A Type II error occurs when we do not reject a null hypothesis that is ______.

false

3) A company performs a statistical test for the alternative hypothesis that the average number of laptops per household in California is not 2. After calculating the P-value, the company fails to reject the null hypothesis. The interpretation of this conclusion is "We ______ evidence that the average number of laptops per household in California ______ 2."

have, is not

Left tail test/Right tail test

if shaded region is to the left of null hypothesis or to the right (i.e., whatever side the z-score is on)

The ___________ hypothesis, denoted H0, is the statement given the benefit of the doubt.

null

The p-value is defined as the probability ______.

of observing a test statistic that is at least as contradictory to the null hypothesis as the test statistic actually computed from a sample

A drug company has developed a new form of insulin intended to keep blood sugar at healthy levels for longer periods than drugs currently on the market. The claim that the new insulin is superior to current drugs is

research hypothesis

Z in test statistic formula comes from what?

sample data

p-value is a _________ area on a bell-curve

shaded

We would like both a and B to be ___________

small

a p-value of more than .05 means

that there is no effect

a p-value of less than .05 means

that we have an effect

In a "greater than" hypothesis test, the null hypothesis is rejected if and only if ______.

the p value is less than a

A Type I error occurs when we reject a null hypothesis that is ______.

true

Test Statistic Formula

z (z-score) = x-bar (sample mean) - mu (hypothesis/population mean)/ (standard deviation/square root of number of samples)

Test statistic formula

z = x−μσ/√n

Margin of error=

z-score x standard deviation