QMS202Jason Yim

Jessica Shute

Normal distribution

Normal distribution question Z-test question

Confidence Intervals

2800 students asked if they’re comfortable to report cheating by their fellow students. 1344 said Yes, 1456 said No. What is the confidence Interval for the proportion of student population who feel comfortable reporting cheating?

Sample size

Sample size determination for the Mean Suppose you want to estimate the population mean to break a

wooden board to within ±20 kilos with 99% confidence . The standard deviation is 80 kilos. Find the sample size!

To find Z: 99% confidenceSTAT, DIST, NORM, INVN Area: 0.005, σ=1, µ=0 EXECUTE!

Implement sample size determination for the mean equation!

n = 106.15

Sample Size Sample size determination for the Proportion

If there is no information available, plug in the following numbers to the equation: e=0.05, π=0.5

Example: In a survey of 500 people, you find that 60% of Facebook users will use the “Like” button on facebook. Determine the sample size needed to estimate the proportion of facebook users who use the “Like” to within ±0.07 with a 94% confidence

To find Z: 99% confidenceSTAT, DIST, NORM, INVN Area: 0.03, σ=1, µ=0 EXECUTE!

N= 173.24

Types of Errors

Type I error If you reject the null hypothesis H0 when it is true

and should not be rejected Probability of Type I error =

Type II error If you do not reject the null hypothesis H0 when it

is false and should be rejected Probability of Type II error =

Hypothesis Testing and Decision Making

Actual Situation

Statistical Decision H0 True H0 False

Do not reject H0Correct decisionConfidence = 1 -

Type II errorP(Type II error) =

Reject H0Type I errorP(Type I error) =

Correct decisionPower = 1 -

When do we conduct one and two-tail tests? A two-tail test is conducted whenever the alternative hypothesis specifies that

the mean is not equal to the value stated in the null hypothesis H0: µ = µ0

H1: µ ≠ µ0

There are two types of one-tail tests: A one-tail test that focuses on the right tail is conducted whenever the

alternative hypothesis states that the mean is greater than the value stated in the null hypothesis H0: µ = µ0

H1: µ > µ0

A one-tail test that focuses on the left tail is conducted whenever the alternative hypothesis states that the mean is less than the value stated in the null hypothesis H0: µ = µ0

H1: µ < µ0

1-Sample Z-test

Assume Normal distribution Example: Boozehead Co. produces 40oz beers, and

they’re suspected of cheating their consumers for their beer. They believe that they’re getting less than 40oz. A bunch of Boozehead beer drinkers approached head office who has found out that the bottling process that fills this type of bottle has a standard deviation of 1.5oz. For evident reasons, a sample of 48 beers were measured, and they found that the mean volume was 39.7oz. With a 95% confidence level, does this evidence support the assumption of Boozehead beer drinkers?

6 Step Method of Hypothesis Testing1. State the null hypothesis H0 and alternative hypothesis H1

2. Choose the level of significance, , and sample size, n

3. Determine the appropriate test statistic and sampling distribution

4. Determine the critical values that divide the rejection and non rejection regions

5. Collect the data and compute the value of the test statistic

6. Make the statistical decision and state the managerial conclusion.

If the test statistic falls into the non-rejection region, you do not reject the null hypothesis.

If the test statistic falls into the rejection region, reject the null hypothesis.

***** Put in sample question

5 Step Method of Hypothesis Testing Using P-Value1. State the null hypothesis, H0, and the alternative hypothesis, H1

2. Choose the level of significance, , and sample size, n

3. Determine the appropriate test statistic and sampling distribution

4. Collect the data, compute the value of the test statistic, and compute the p-value

5. Make the statistical decision and state the managerial conclusion

If the p-value ≥ , you do not reject the null hypothesis H0

If the p-value < , reject the null hypothesis H0

Recall: if p-value is low, H0 must go

***** Put in sample question from textbook

One-Tail Tests A company that manufactures Basketballs is concerned that the

mean weight doesn’t exceed 20 ounces. Standard deviation is 0.04 ounces. A sample of 40 basketballs were selected and the sample mean is 20.12 ounces. With 99% confidence level, is there evidenec that the population mean weight of the basketballs is greater than 20 ounces?

Matched Pairs t-test Can more men get into Night Clubs when

they bring ladies in with them? To investigate this possibility, a random sample of 6 clubs are observed at the line-up to see how many men get into the club with ladies, and without ladies.

At 0.01 level of significance, is there evidence of that the men admitted to clubs between men coming in alone greater than men bringing in women to clubs?

Is p-value >0.01? Step 1: Enter a list of the difference

between the numbers.You can solve for two ways! P-value or T-test Step 2: t_calc = t_6-1, 0.01= -3.3649, t_crit

= -1.6682, We reject Ho! Step 2: p-value >0.01? Yes! Do not reject

Ho! There is enough evidence to conclude that

there aren’t more men coming in alone than men bringing in women to clubs

Clubs Men Men bringing in women to the club

Circa 270 483

Republik 352 389

Lot122 145 150

Guv 156 400

Mckey Finns

206 278

Revival 216 145

1-Sample t Test

Unknown standard deviation Boozehead Co. produces beer bottles that hold 8oz of beer.

Boozehead…. AGAIN gets complaints that their consumers get less than 8oz. They sample 22 bottles and finds the average amount of liquid held by bottles is 7.802 with a std dev of 0.302. Is there evidence that Boozehead Co. offers their consumers less than 8oz of beer?

2-sample t-test Comparing the means of two

independent populations Jack and Jill decided to play

Paper Toss, to see who can throw in more paper balls into the garbage can within a certain time. Jack and Jill did 10 trials and ended up with the following results. At the 0.05 level of significance, is there evidence that the Paper Toss mean is lower for Jack than it is for Jill?

Ho: µ

Jack Jill

6 5

8 7

3 9

2 2

6 3

4 5

1 2

8 6

9 3

8 1

1 sample z-test proportion A new vending machine was put into the Ted Rogers Business

Building to make sure that it wasn’t eating people’s money. The previous vending machine was successful to complete transactions 85% of the time. A sample of 200 vending machine transactions were observed, and found that 195 orders completed the transaction successfully. With the 0.02 level of significance, can you conclude that the new vending machine has increased the proportion of successful completed transactions?

Ho: π = 0.85 (the proportion of successfully completed orders is less or equal to 0.85)

H_1: π >0.85 ( the proportion of orders filled correctly is greater than 0.85)

2-sample Z-test A company makes cigarettes out of two machines. There have

been complains that the some cigarettes were shorter in length and getting ripped off for what they were paying. In the past, it has been determined by past studies that the standard deviation of the cigarette length is 1.5cm from Machine 1, and 1.9 from Machine 2. The manufacturer decided to take samples of the cigarettes from both machines to test to see weather the mean length of the cigarettes from machine 2 was significantly larger than the mean length in machine 1. The sample of 100 cigarettes from Machine 1 had a mean length of 6.94cm, and Machine 2 had a mean length of 6.92. At 10% level of significance what is the conclusion?

F-test for the difference between 2 variances

Finding Lower-Tail Critical Values Numerator and denominator for F_U and F_L

are switched around!

F-test for the difference between 2 variances

A engineering team has developed a new machine to assemble car engines, and they want to know if this new machine has reduced the time to assemble the car engine. They have taken samples of 60 engines from the new machines, and 50 engines from the old machine. The mean is 230 minutes, and the standard deviation is 4 minutes for the new machines. For the old machine, the mean is 226 minutes with a standard deviation of 6 minutes. They want to test to see if the assumption about equal variances seems reasonable at the 5% level.

One-way ANOVA

At the Italian restaurant, Westside Marios, a study was done to see if there is evidence whether there is a difference in mean time to serve food from the following 4 entrees.

At the 0.05 level of significance, is there evidence of a difference in the mean time to serve food from the 4 entrees?

Entrees Minutes

Gnocchi 10

Ravioli 12

Spaghetti 7

Linguine 9

Sample Midterm1 The standard error of the mean

a) is never larger than the standard deviation of the populationb) measure the variability of the mean from sample to samplec) decreases as sample size increasesd) all of the above

2 For air travelers, one of the biggest complaints is from the waiting time between when the airplane taxis away from the terminal until the time it takes off. This waiting time is known to have a skewed-right distribution with a mean of 10 minutes and a standard deviation of 8 minutes. Suppose 100 flights have been randomly sampled. Describe the sampling distribution of the mean waiting time between when the airplane taxis away from the terminal until the flight takes off for these 100 flights.

a) Distribution is skewed-right with mean = 10 minutes and standard error = 8 minutesb) Distribution is approximately normal with mean = 10 minutes and standard error = 0.8 minutesc) Distribution is skewed-right with mean = 10 minutes and standard error = 0.8 minutesd) Distribution is approximately normal with mean = 10 minutes and standard error = 8 minutes

3 If the standard error of the sampling distribution of the sample proportion is 0.0229 for samples of size 400, then the population proportion must be either:

a) 0.4 or 0.6b) 0.5 or 0.5c) 0.3 or 0.7d) 0.2 or 0.8

4 When determining the sample size for a proportion for a given level of confidence and sampling error, the closer to 0.50 that π is estimated to be, the sample size required

a) is not affected

b) is larger

c) is smaller

d) can be smaller, larger or unaffected

5 A 99% confidence interval estimate can be interpreted to mean thata) if all possible samples are taken and confidence interval estimates are developed 99% of them would include the true

population mean somewhere within their interval

b) we have 99% confidence that we have selected a sample whose interval does include the population mean

c) both of the above

d) none of the above

6 If you were constructing a 99% confidence interval of the population mean based on a sample of n=25, where the standard deviation of the sample s = 0.05, what will the critical value of t be?

a) 2.7874

b) 2.4851

c) 2.7969

d) 2.4922

a) A confidence interval ws used to estimate the proportion of statistics students that are females. A random sample of 72 statistics students generated the following 90% confidence interval: (0.438, 0.642). Based on the interval above, is the population proportion of females equal to 0.60?

a) Maybe, 0.60 is a believable value of the population proportion based on the information above

b) No and we are 90% sure of it

c) No, the population is 54.17%

d) Yes and we are 90% sure of it

8 As an aid to the establishment of personnel requirements, the director of a hospital wishes to estimate the mean number of people who are admitted to the emergency room during a 24-hour period. The director randomly selects 64 different 24-hour periods and determines the number of admissions for each. For this sample, mean = 19.8 and s2 = 25. Using the sample standard deviation as an estimate for the population standard deviation, what size sample should the director choose if she wishes to estimate the mean number of admissions per 24-hour period to within 1 admission with 98% reliability?

a) 106

b) 135

c) 136

d) 105

9 A university dean is interested in determining the proportion of students who receive some sort of financial aid. Rather than examine the records for all students, the dean randomly selects 200 students and finds that 118 of them are receiving financial aid. If the dean wanted to estimate the proportion of all students receiving financial aid to within 3% with 99% reliability, how many students would need to be sampled?

a) n = 1,843

b) n = 1

c) n = 1,504

d) n = 1,784

10 True/False: The t distribution approaches the standardized normal distribution when the number of degrees of freedom increases.

11 True/False: The managers of a company are worried about the morale of their employees. In order to determine if a problem in this area exists, they decide to evaluate the attitudes of their employees with a standardized test. They select the Fortunato test of job satisfaction which has a known standard deviation of 24 points.

Referring to the above information, this confidence interval is only valid if the scores on the Fortunato test are normally distributed.

12 A hotel chain wants to estimate the average number of rooms rented daily in each month. The population of rooms rented daily is assumed to be normally distributed for each month with a standard deviation of 24 rooms.

Referring to the above information, during January, a sample of 16 days has a sample mean of 48 rooms. This information is used t o calculate an interval estimate for the population mean to be from 40 to 56 rooms. What is the level of confidence of this interval?

a) 95.0%

b) 81.8%

c) 49.5%

d) 26.1%

13 A university wanted to find out the percentage of students who felt comfortable reporting cheating by their fellow students. A surveyed of 2,800 students was conducted and the students were asked if they felt comfortable reporting cheating by their fellow students. The results were 1,344 answered “yes” and 1,456 answered “no”.

Referring to the above information, a 99% confidence interval for the proportion of student population who feel comfortable reporting cheating by their fellow students of from _____ to _____.

a) 49.6% to 54.4%

b) 45.6% to 50.4%

c) 46.1% to 49.9%

d) 50.1% to 53.9%

14 A major DVD rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area are equipped with DVD players. It conducts a telephone poll of 300 randomly selected households in the area and finds 96 have DVD players. State the test of interest to the rental chain.

a) Ho: π = 0.25 versus H1: π > 0.25 c) H0: π = 5,000 versus H1: π > 5,000

b) H0: π = 0.32 versus H1: π > 0.32 d) H0: µ = 5,000 versus H1: µ > 5,000

15 A major DVD rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area are equipped with DVD players. It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have DVD players. The p-value associated with the test statistic in this problem is approximately equal to

a) 0.0100 c) 0.0051

b) 0.0026 d) 0.0013

16 An entrepreneur is considering the purchase of a coin-operated laundry machine. The current owner claims that over the past 5 years, the average daily revenue was $675 with a standard deviation of $75. A sample of 30 days reveals a daily average revenue of $625. If you were to test the null hypothesis that the daily average revenue was $675, which test would you use?

a) t-test of a population proportion c) Z-test of a population proportion

b) Z-test of a population mean d) t-test of a population mean

17 An entrepreneur is considering the purchase of a coin-operated laundry machine. The current owner claims that over the past 5 years, the average daily revenue was $675 with a standard deviation of $75. A sample of 30 days reveals a daily average revenue of $625. If you were to test the null hypothesis that the daily average revenue was $675 and decide not to reject the null hypothesis, what would you conclude?

a) There is not enough evidence to conclude that the daily average revenue was not $675

b) There is enough evidence to conclude that the daily average revenue was $675

c) There is enough evidence to conclude that the daily average revenue was not $675

d) There is not enough evidence to conclude that the daily average revenue was $675

18 True/False: The test statistic measures how close the computed sample statistic has come to the hypothesized population.

19 True/False: Suppose we wish to test H0: µ = 8 versus H1 < 8. What will result if we conclude that the mean is less than 8 when its true value is really 12?

a) We have made a Type I error

b) We have made a Type II error

c) We have made the correct decision

d) We do not have enough information

20 True/False: Suppose we wish to test H0: π = 75% versus H1 > 75%. What will result if we conclude that the proportion is not greater than 75% when its true value is really 85%?

19 We have made a Type II error

20 We have made a Type I error

21 We do not have enough information

22 We have made the correct decision

Fill in the following table for the next 2 questions:

Null: Assumptions:

Alternative:

Test Statistic

Comparison

Conclusion

21 A major home improvement store conducted its biggest brand recognition campaign in the company’s history. A series of new television advertisements featuring well-known entertainers and sports figures were launched. A key metric for the success of television advertisements is the proportion ofviewers who, “like the ads a lot.” A study of 1,189 adults who viewed the ads reported that 230 indicatedthat they, “like the ads a lot.” The percentage of a typical television advertisement receiving the, “likethe ads a lot,” score is believed to be 22%. Company officials wanted to know if there is evidence that the series of television advertisements are less successful than the typical ad at a 3% level of significance.

Use the above information and the appropriate set of hypothesis testing to determine there is evidencethat the series of television advertisements are less successful than the typical ad at 3% level of significance. State any necessary assumptions.

THINGS TO REMEMBER Know when standard deviation is known or unknown to use the

appropriate testing (Z or t?, sample standard deviation or not?) Recognize the difference between matched t-test (independent or

dependent) Easy to evaluate p-value! If p-value is greater than its level of

significance, we DO NOT REJECT H_o, if p-value is less than its level of significance, we REJECT Ho!

Draw out charts to see where your t, z, and f values are! Write in assumptions for easy marks Don’t confuse null hypothesis, and alternative hypothesis Always follow the 5 steps! Understsand calculator implementations and what they all mean!

GOOD LUCK!

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