STAT 202 - EXAM 1

Means (average)

- average quality level of products, suppliers, etc. - average time to complete task - average sales, # of products sold -average # of safety incidents in a warehouse or manufacturing plant

paired sample tests of means

- overtime - after use - after an intervention - raters of same or different items - DEPENDENT EXAMPLE: observe the blood pressure of patients before giving a stimulus and again observe the blood pressure of the same patient after the stimulus has given t-calc = dbar -mu(d)/ (s(d)/√n)

proportions

- products is defective or not defective, supplier is satisfactory or unsatisfactory - completing a task in under a certain amount of time vs. taking longer (meeting or missing benchmark) - selling more or less products than a competitor, hitting vs. missing dollar target - rate of safety incidents

interactive effects (3 hypotheses)

HYPOTHESES: Factor A (clinics) H0: A1=A2=A3=A4 H1: not all clinic delivery times are equal Degrees of freedom: 4-1=3 Factor B (suppliers) H0: B1=B2=B3=B4=B5 H1: not all supplier delivery times are equal Degrees of freedom: 5-1=4 Interactive effect (clinics x suppliers) H0: all ABjk are equal H1: not all interactive effects are equal Degrees of freedom: (5-1)(4-1)=12 Find F.INV for all 3 critical value Conclude

A company has conducted a statistical two sample hypothesis test of means with known variances with one tail. The critical value they used was +1.645. The test statistic they calculated was 1.730. Based on this, they decided to reject the null hypothesis. What can you say about their sensitivity to their choice of alpha? If they had selected a larger alpha, they would have made the decision to fail to reject. If they had selected a smaller alpha, they would have made the decision to fail to reject. Their decision to reject is not sensitive to their choice of alpha None of the above

If they had selected a smaller alpha, they would have made the decision to fail to reject

An independent consumer review company wants to examine average battery life for smartphones to see if differences exist based on the brand or the age of the smartphone. They would like to compare the iPhone, Google Pixel and Samsung Galaxy and will look at phones that are brand new, 6 months old and 1 year old. To conduct a statistical test, they measured the battery life of 25 smartphones for each brand/age combination. Which of the following tests should they conduct?

Two factor ANOVA with replication

two-factor ANOVA

What if the retailer also buys paint from 3 different suppliers and is considering that supplier ALSO impacts paint viscosity (not just temperature). Now we have 2 factors - each with 3 categories

Which of the following statements about ANOVA is true?

When SSE (unexplained variance) is sufficiently large relative to SSB (explained variance), we reject the null hypothesis.

ANOVA

a test used to detect sources of variation in a numerical dependent variable -Only compares means

a large company owns 4 manufacturing plants. at each of the 4 plants, they collected a sample of 100 products and for each product, they determined whether or not the product was defective. the company would like to know if the rate of defects significantly differs across the 4 manufacturing plants. they can use ANOVA to test their hypotheses.

false

when I have more than two samples of proportions, I can use anova to test for significant differences

false

type II

false negative

type I

false positive

Deciding on means test

if a benchmark/population - use single sample hypothesis test Is there standard deviation of the population? Yes - *known standard dev.* no - *unknown standard dev* if another sample - use a two sample hypothesis test do I know the standard deviation of the population for each sample? yes - test w/ *known standard deviation* no - do i have reason to believe that the standard dev. are the sample/ am I okay with the less conservative test? yes - test with unknown but equal standard dev. no - unknown but unequal standard dev.

Tukey's test should be used only if

the overall null hypothesis is rejected

A university wants to start placing coupons on admission tickets to basketball games in an effort to increase concession sales. They decide to test different versions of coupons. Over the course of the next basketball season, they will place a coupon for popcorn on the back of tickets at 10 games, a coupon for soda on the back of tickets at 10 games, and a coupon for hotdogs on the back of tickets at 10 games. They will record concession sales at each game and calculate an average of concession sales for each coupon type. They want to determine whether or not coupon type leads to significant differences in concession sales. They must use ANOVA to test this.

true

I can only test for interaction effects with ANOVA when I have replication.

true

Two Factor ANOVA with replication results in 3 decisions to reject or fail to reject

true

When conducting a single sample hypothesis test of means or proportions, I am comparing the mean or proportion of a sample to a benchmark mean or proportion.

true

When presented with sample means for more than two samples, I cannot conduct pairs of two sample hypothesis tests because this increases the risk of Type 1 error.

true

by increasing alpha, we increase the probability of a Type 1 error.

true

Which of the following statements is NOT true regarding ANOVA?

Tukey's Test can be conducted after running a Two Factor ANOVA without replication

In single or two sample hypothesis testing of means, when I do not know the variance (standard deviation), I use the F-distribution.

false

When I have more than two samples of proportions, I can use ANOVA to test for significant differences.

false

When presented with sample means for more than two samples, I cannot conduct pairs of two sample hypothesis tests because this increases the risk of Type II error.

false

steps for single means

1) H0: mu0 => =< = to mu1 H1: mu0 > < to mu1 2) Determine if left or right, two tailed test 3) critical value (0.1, 0.05, 0.01) 4) Calculate the test stat *KNOWN = sample mean - mu0 / (stdev/√n)* *UNKNOWN = xbar - mu0/ (s/√n)* 5) compare test stat to c.v 6) reject or no reject

steps for 2 tailed w/ variance

1) H0: mu1(#) - mu2(#) = 0 >= or <= H1: mu1(#) - mu2(#) =/ 0 < or > 2) critical value (0.1, 0.05, 0.01) 3) Calculate the test stat *z calc = ((xbar1 - xbar2) - (mu1-mu2))/ √(o1^2/n1) + (o2^2/n2)* 4) compare test stat to c.v 5) reject or no reject

Steps for single Proportions

1) H0: pi0 => =< = to pi1 H1: pi0 > < to pi1 2) Determine if left or right, two tailed test 3) critical value (0.1, 0.05, 0.01) 4) Calculate the test stat z calc = p - pi0/ (√pi0 (1-pi0)/n) 5) compare test stat to c.v 6) reject or no reject

two tailed unknown variance (assumed equal)

1) define the null and alternate hypotheses H0: mu1(#) - mu2(#) = 0 H1: mu1(#) - mu2(#) =/ 0 2) Find C.V. using degrees of freedom dof = n1+n2-2 n = sample size 3) collect data and calculate T-stat *t-calc = (xbar1 - xbar2) - (mu1 - mu2)/√(sp^2/n1 + sp^2/n2) *pooled variance* or no? 4) make a decision by looking at t-stat and c.v.

two tailed unknown variance (assumed unequal)

1) define the null and alternate hypothesis 2) critical value using DOF *find formula* 3) t stat t-calc = (xbar1 - xbar2) - (mu1 - mu2) / √(s1^2/n1) + (s2^2/n2) 4) reject or no?

In a 15-day survey of air pollution in two European capitals, the mean particulate count in Athens was 39.5 with a standard deviation of 3.75 while in London the mean was 31.5 with a standard deviation of 3.35. Which of the following is the most appropriate test to conduct to determine if the pollution level is significantly higher in Athens? A two-tailed, two sample hypothesis test of means with unknown variance but assumed equal A one-tailed, two sample hypothesis test of means with unknown variance but assumed equal A one-tailed, two sample hypothesis test of means with known variance A two sample hypothesis test of paired means

A one-tailed, two sample hypothesis test of means with unknown variance but assumed equal

Paired t-test

A test designed to determine the statistical difference between two groups' means where the participants in each group are either the same or matched pairs. Hypothesis H0: md (-1.6) = 0 H1: md (-1.6) =/ 0

Tukey's Test (ANOVA)

Can be used to compare the difference in means between each sample. Means that do not share a letter a significantly different

ANOVA numbers

Critical Value = F Test Statistic = F Crit

When conducting a two sample hypothesis test of means or proportions, I am comparing the mean or proportion of a sample to a benchmark mean or proportion. (T/F)

False

ANOVA steps

H0: mu1=mu2=mu3=mu4 (all means are the same) H1 = Not all means are the equal --> YOU MUST SAY AT LEAST ONE IS DIFFERENT Find Critical Values We use F-Distribution F.INV.RT(X,X,X) Find T-stat: F= MSB/MSE = (SSB/c-1)/(SSE/n-c) Tukeys Test - to find out which observations diff Only use tukey test when there are more than one observation from each combo

A company has conducted a statistical two sample hypothesis test of means with known variances with one tail. The critical value they used was +1.645 (alpha of 0.05). The test statistic they calculated was 1.730 (pvalue of 0.042). Based on this, they decided to reject the null hypothesis. What can you say about their sensitivity to their choice of alpha? If they had selected a larger alpha, they would have made the decision to fail to reject. If they had selected a smaller alpha, they would have made the decision to fail to reject. Their decision to reject is not sensitive to their choice of alpha None of the above

If they had selected a smaller alpha, they would have made the decision to fail to reject.

The average miles per gallon (mpg) for a 2009 Toyota Prius was 45.5 with a standard deviation of 1.8 in a sample of 10 tanks of gas. The average mpg for a 2009 Honda Insight was 42.0 with a standard deviation of 2.3 in a sample of 10 tanks of gas. We would like to know if the Toyota Prius has a different average mpg than a Honda Insight. Which of the following is the correct set of hypotheses for this test? Null Hypothesis: mu1 (mpg Toyota)- mu2 (mpg Honda) = 0 Alternative Hypothesis: mu1 (mpg Toyota) - mu2 (mpg Honda) \= 0 * \= means "not equal to" Null Hypothesis: mu1 (mpg Toyota)- mu2 (mpg Honda) <= 0 Alternative Hypothesis: mu1 (mpg Toyota) - mu2 (mpg Honda) > 0 Null Hypothesis: mu1 (mpg Toyota)- mu2 (mpg Honda) >= 0 Alternative Hypothesis: mu1 (mpg Toyota) - mu2 (mpg Honda) < 0 None of the above

Null Hypothesis: mu1 (mpg Toyota)- mu2 (mpg Honda) = 0 Alternative Hypothesis: mu1 (mpg Toyota) - mu2 (mpg Honda) \= 0 * \= means "not equal to"

A 20-minute consumer survey that included a $5 Starbucks gift card was emailed to 500 adults aged 25-34. The survey resulted in 65 responses. The same 20-minute consumer survey was emailed to 500 adults aged 25-34 without the Starbucks gift card. The survey resulted in 45 responses. We would like to know if the inclusion of a Starbucks gift card led to an increase in survey response rate. Which of the following hypotheses should be used? Null Hypothesis: mu1 (rate with Starbucks gift card)<= mu0 (rate without Starbucks gift card) Alternative Hypothesis: mu1 (rate with Starbucks gift card)> mu0 (rate without Starbucks gift card) Null Hypothesis: mu1 (rate with Starbucks gift card)= mu0 (rate without Starbucks gift card) Alternative Hypothesis: mu1 (rate with Starbucks gift card) \= mu0 (rate without Starbucks gift card) * \= means "not equal to" Null Hypothesis: pi1 (rate with Starbucks gift card)= pi2 (rate without Starbucks gift card) Alternative Hypothesis: pi1 (rate with Starbucks gift card) \= pi2 (rate without Starbucks gift card) * \= means "not equal to" Null Hypothesis: pi1 (rate with Starbucks gift card) - pi2 (rate without Starbucks gift card)<=0 Alternative Hypothesis: pi1 (rate with Starbucks gift card) - pi2 (rate without Starbucks gift card) > 0

Null Hypothesis: pi1 (rate with Starbucks gift card) - pi2 (rate without Starbucks gift card)<=0 Alternative Hypothesis: pi1 (rate with Starbucks gift card) - pi2 (rate without Starbucks gift card) > 0

The U.S. government program "cash for clunkers" encourages individuals to trade in their old gas-guzzling cars for vehicles that get noticeable better gas mileage - where noticeably is more than 5 mpg increase. in a random sample of 14 participants in the program. it was found that the average mpg of individual's old cars was 15.6 with a standard deviation of 6.4 and the average mpg of individuals new cars was 23.8 with a standard deviation of 9.2. we would like to know if there is insufficient evidence to show that the governments program is providing the stated benefit. which of the following sets of hypotheses should be used to test this?

Null hypothesis:- mu1(new car)-mu2(old car)<=5 Alternative hypothesis:- mu1(new car) - mu2( old car)>5

The Scottsdale fire department aims to respond to fire calls in 4 minutes or less on average. Historically, response times are normally distributed with a standard deviation of 1 minute. A random sample of 18 fire calls showed a mean response time of 4 minutes 30 seconds. The department would like to know if the sample provides evidence that the mean response time goal is not being met. Which test should be used? Single sample hypothesis test of means with known variance (standard deviation) Single sample hypothesis test of means with known variance (standard deviation) Single sample hypothesis test of means with unknown variance (standard deviation) Two sample hypothesis test of means with known variance (standard deviation) Single sample hypothesis test of proportions

Single sample hypothesis test of means with known variance (standard deviation)

According to J.D. Power & Associates, the mean purchase price of a smartphone device (such as an iPhone or Android) in 2010 was $216. In 2011, a random sample of 20 business managers who owned a smartphone device showed a mean purchase price of $209 with a sample standard deviation of $13. We would like to conclude whether or not the average price of a smartphone changed from 2010 to 2011. What test should be used? Single sample hypothesis test of means with known variance (standard deviation) Single sample hypothesis test of means with unknown variance (standard deviation) Two sample hypothesis test of means with known variance (standard deviation) Two sample hypothesis test of means with unknown variance (standard deviation) but assumed equal

Single sample hypothesis test of means with unknown variance (standard deviation)

To encourage efficiency at a drive through window, a fast food restaurant chain specifies, as a guideline, that at least half (50%) of all drive through orders should be taken, prepared, and delivered in 3 minutes or less. Subsequently, a random sample of 64 drive through orders was taken and it was found that 24 of them were taken, prepared and delivered in 3 minutes or less. The remaining orders from the sample took longer than 3 minutes. The fast food company would like to know if they are meeting the specified guideline. Which test should be used? Single sample test of means with unknown variance (standard deviation) Two sample test of proportions Single sample test of proportions Two sample test of means with unknown variance (standard deviation)

Single sample test of proportions

A large company owns 4 manufacturing plants. At each of the 4 plants, they collected a sample of 100 products and for each product, they determined whether or not the product was defective. The company would like to know if the rate of defects significantly differs across the 4 manufacturing plants. They can use ANOVA to test their hypothesis.

false

In hypothesis testing of means, when I do not know the variance (standard deviation), my test statistic is found using the Student T-distribution. (T/F)

True

I have two paired samples of means and should conduct a two sample hypothesis test of paired means with unknown variance. If I instead conduct a two sample test of independent means with unknown variance, I am more likely to fail to reject the null hypothesis. (T/F)

True

An executive at a firm found that the mean length of a sample of 64 of her own telephone calls during July was 4.48 minutes with a standard deviation of 25.81. She decided to make an effort to reduce the length of her telephone calls. In August, a sample of 48 of her own telephone calls showed a mean of 2.39 with a standard deviation of 2.018 minutes. Which of the following tests is most appropriate to determine whether or not she decreased her average telephone call time? Two sample hypothesis test of paired means Two sample hypothesis test of means with known variance Two sample hypothesis test of means with unknown variance assumed equal Two sample hypothesis test of means with unknown variance assumed unequal

Two sample hypothesis test of means with unknown variance assumed unequal

In an October issue of Muscle and Fitness, there were 252 ads with 97 of them being full-page. For the same month, the magazine Glamour had 342 ads with 167 being full-page. You would like to know if the two magazines differ in the level of full-page ads that they run. Which of the following is the most appropriate test to use? Single sample hypothesis test of means with known variances Single sample hypothesis test of proportions Two sample hypothesis test of proportions Two sample hypothesis test of paired proportions

Two sample hypothesis test of proportions

A restaurant/bar conducted random sampling of bar purchases. In a sample of 17 restaurant patrons, they found that, when the background music tempo was slow, the mean purchase amount was $30.47 with a standard deviation of $15.10. In a sample of 14 patrons in the same restaurant, they found that, when the background music tempo was fast, the mean purchase amount was $21.62 with a standard deviation of $14.80. The bar would like to know if they can conclude that background music tempo leads to different average purchase amounts at the bar. What test should be used? Single sample test of means with known variance (standard deviation) Two sample test of means with known variance (standard deviation) Two sample test of means with unknown variance (standard deviation) but assumed equal Two sample test of means with unknown variance (standard deviation) but assumed unequal

Two sample test of means with unknown variance (standard deviation) but assumed equal

A grocery store was interested in understanding the level of brand loyalty for different products. The store randomly polled 100 customers who had just purchased milk and found that, of the 100 customers, 53 were loyal to one brand of milk. The store then randomly polled 100 customers who had just purchased bath soap and found that, of the 100 customers, 65 were loyal to one brand of soap. The grocery store would like to know if this shows evidence that customers display stronger brand loyalty to soap than milk. What test should be used? Two sample test of means with unknown variance (standard deviation) but assumed equal Two sample test of proportions Two sample test of paired means with unknown variance (standard deviation) Two sample test of means with unknown variance (standard deviation) but assumed unequal

Two sample test of proportions

the manufacturer of an airport baggage scanning machine claims that it can handle an average of 530 bag per hour. an independent party does a random test of 16 hours and found that the average in the sample was 510 bags with a standard deviation of 50. a statistical hypothesis test was conducted to determine if the average bags per hour is less than the stated amount. which of the following statements regarding errors is true?

Type 1 error is finding that average is less than 530 when in reality the machine can handle at least 530; Type 2 error is finding average is at least 530 when in reality the machine cannot handle 530

Consider the hospital from problem 4 of the homework that is using a statistical test to decide whether or not to accept future shipments of syringes from a supplier. The two errors the can make are to reject shipments that are actually acceptable and to accept shipments that do not meet the requirements. The hospital has decided that the error of accepting shipments that do not meet the requirements is substantially more important. They would like to ensure that they do not make this error even if that means increasing the chance that they reject acceptable shipments. When selecting their alpha (choosing between 0.01, 0.05 and 0.10) which alpha level provides the highest protection against this type of error? alpha=0.01 alpha=0.05 alpha=0.10

alpha=0.10

Which of the following distributions is used to find the critical value and test statistic for ANOVA?

f-distribution

A hospital is investigating the average length of stay (LOS) of patients who arrive at the emergency department with different fractures. They would like to know if the average LOS for patients with facial fractures is significantly different from the average length of stay of patients with femur/hip fractures. The hospital has collected LOS for 20 patients with facial fractures and 20 patients with femur/hip fractures. To test their hypothesis, they MUST use ANOVA.

false

A university business school is interested in the difference of GPAs across majors. They collect a sample of 7 students from each of 4 different majors: accounting, finance, marketing and analytics. They found that the average GPA was 3.0 for accounting, 3.6 for finance, 3.1 for marketing and 3.5 for analytics. After conducting an ANOVA, the university rejected the null hypothesis. Can they conclude that the GPA of finance students is significantly different from accounting students?

no

one sample vs. two sample

one sample - compared to benchmark two sample - literally compared to another sample taken/given

1. A production facility has a constant flow of truck drivers that pickup products to deliver to customers. Recently, they have installed a new automatic gate system. Occasionally, truck drivers have issues either entering or exiting the facility using the automatic gate system. Over the course of one week, the production facility randomly selected 8 truck drivers. The random sample showed an average of 4.5 (with a standard deviation of 3.3) issues with entry into the facility and an average of 6.6 (with a standard deviation of 2.0) issues with exit from the facility. The production facility would like to know if the new automatic gate system has more issues with exiting the facility than entering it. What test should be used?

two sample tests of paired means

Unequal variances vs equal variances

unequal = greater than 20% apart equal = less than 20% apart

A university business school is interested in the difference of GPAs across majors. They collect a sample of 7 students from each of 4 different majors: accounting, finance, marketing and analytics. They found that the average GPA was 3.0 for accounting, 3.6 for finance, 3.1 for marketing and 3.5 for analytics. After conducting an ANOVA, the university rejected the null hypothesis. Can they conclude that the GPA of students varies across the 4 majors?

yes