. A standard gold bar weighs exactly 12.4 kg. We have two scales of unknown accuracy and precision. In particular, unknown to us, using Scale #1, measurements of the bar’s weight act like random draws from a N(12.2, 0.05) distribution while, using Scale #2, measurements of the bar’s weight act like random draws from a N(12.4, 0.2) distribution. Using each scale, we take 10 measurements and compute a 95% confidence interval for the true weight of the bar. Neither includes 12.4 kg.

a. What is the best explanation for why the confidence interval based on the Scale #1 measurements does not contain 12.4 kg?

b. What is the best explanation for why the confidence interval based on the Scale #2 measurements does not contain 12.4 kg?

Problem #2: Does presentation methodology improve retention for remembered material? You know that the average number of concepts remembered from a lecture is 20. You present information using a song to a group of four participants. Here are their scores for the number of concepts they remember from a lecture: 20, 21, 18, 29 Please conduct all steps of hypothesis testing and evaluate using a one tailed test and alpha =.01. You must use all five steps in hypothesis testing:

Restate the question as a research hypothesis and a null hypothesis about the populations.

Determine the characteristics of the comparison distribution.

Determine the cutoff sample score on the comparison distribution at which the null hypothesis should be rejected.

Determine your sample’s score on the comparison distribution.

Decide whether to reject the null hypothesis.

The Manager at Rainbow Valley II advertises that the typical family visiting the park spends at least one hour in the park during weekends. A sample of 25 visitors during the weekends in the month of July revealed that the mean time spent in the Park was 63 minutes with a standard deviation of 8 minutes.

1. Using the 0.01 significance level and a one-tailed test, is it reasonable to conclude that the mean time in the Park is greater than 60 minutes? Show all steps in your test of hypothesis.

2. Repeat the analysis at the 0.05 significance level. (You may show only the calculations that change).

3. Repeat with a survey mean of 64 minutes at the .01 significance level. (You may show only the calculations that change).

4. What do you conclude from your analysis?

Note: If the alpha level is not included, set the alpha to .05. Problem 1: The Graded Naming Test (GNT) asks respondents to name objects in a set of 30 black and white drawings in order to detect brain damage. The GNT population norm for adults in England is 20.4. Researchers wondered whether a sample for Canadian adults had different scores from adults in England (Roberts, 2003). If the scores were different, the English norms would not be valid for use in Canada. The mean for 30 Canadian adults was 17.5. Assume that the standard deviation of the adults in England is 3.2. How can we calculate a 95% Confidence Interval (CI) for these data?

Calculate a 95% CI and a 90% CI for this data.

Are the English norms valid for use in Canadian use?

Explain how you know your answer to be true.

How do the two CI’s (95% and 90%) compare to one another?

What is the effect size for these data?

What does this effect size indicate about the meaningfulness of this test for Canadians?

What would you recommend doing to increase the power of this experiment?

PEI Real Estate company believes that their average house price in 2020 of $290,000 is higher than the mean price of all houses sold in 2019 of $270,000. Assuming that their estimate was based on the first 40 sales in 2020 and the population standard deviation of $70,000, test this hypothesis at the 95% confidence interval.

1. State the hypotheses based on a one-tailed test.

2. What is the level of significance?

3. Select a test statistic.  

4. Formulate the decision rule. Sketch this on a graph.

5. Calculate the test value.

6. What is the decision?

7. Does your conclusion change if the 90% confidence interval is selected? Explain.

A manufacturer wants to compare the number of defects on the day shift with the number on the evening shift. A sample of production from recent shifts showed the following defects:

Day Shift 5 8 7 6 9 7

Evening Shift 8 10 7 11 9 12 14 9

The objective is to determine whether the mean number of defects on the night shift is greater than the mean number on the day shift at the 95% confidence level.

1. State the null and alternate hypotheses.

2. What is the level of significance?

3. What is the test statistic?

4. What is the decision rule?

5. Use the Excel Data Analysis pack to analyze the problem. Include the output with your answer. (Note: You may calculate by hand if you prefer).

6. What is your conclusion? Explain.

7. Does the decision change at the 99% confidence level?

For a random sample of 50 measurements of the breaking strength of cotton threads resulting in a mean of 210 grams and standard deviation of 18 grams.

A. Calculate an 80% confidence interval for the true mean breaking strength of cotton threads.

B. Is your interval statistically valid? Explain.

C. Without recalculating, would a 90% confidence result in a wider or narrower interval? Explain.

D. A co-worker offers the following partial interpretation of your interval: We are confident that 80% of all breaking strength measurements of cotton threads will be within this calculated confidence interval. Explain why this interpretation is not correct.

E. Provide an interpretation of your confidence interval.

F.Use R to find the confidence interval you found by hand in the question. Include a copy of your interval and the R commands.

1. Suppose that the average speed of cars driving down Cass avenue is normally distributed with mean 29mph and standard deviation 3mph. a. Find the probability that a car is driving at a speed slower than 26mph or greater than 32mph down Cass avenue. b. Find the probability that a car is driving at a speed between 23mph and 35mph down Cass avenue.  

The brand manager for a brand of toothpaste must plan a campaign designed to increase brand recognition. He wants to first determine the percentage of adults who have heard of the brand. How many adults must he survey in order to be 90% confident that his estimate is within 6 percentage points of the true population percentage.

a. Assume that nothing is known about the percentage of adults who have heard of the brand.

n=

b. Assume that a recent survey suggests that about 79% has heard about the brand.

n=

Read the article titled “What Is the Importance of Probability Rules in a Business?” listed below.

__________________________________________________________________________

What Is the Importance of Probability Rules in a Business?

by Kevin Johnston

Probability has a popular meaning that is not the same as the mathematical meaning. As a small-business owner, you may act on hunches, guesses and instincts. After such actions, you might even say you thought a certain outcome was "probable." However, the mathematics of probability has rules that you can use in a much more disciplined way than guesswork to predict possible outcomes for your business plans.

Classical Approach

The classical approach to using probability depends on several future events that are equally likely to happen. In rolling a die, for example, the odds are equally likely for rolling a 1, 2, 3, 4, 5 or 6. If you roll the die once, you have a 1 in 6 chance of getting the number you want. The formula is the number of favorable outcomes divided by the total number of possible outcomes. Note that if you roll the die twice, the odds are 2 in 12 that you will get the number you want (this is the same value as 1/6). This is because the possible outcomes double if you throw the die twice.

Using the Classical Approach in Business

You can use the classical approach to probability when making business decisions where you don't know the likelihood of several possible outcomes. You assume they are all equally likely, then look at how many attempts you will be able to make. However, in your business, if 6 possible outcomes are equally likely, but they are not affected by how many times you try, you can cut your odds in half with repeated effort. For example, if you make 2 tries, your effort will have a 2 in 6 chance. Notice that 2/6 = 1/3. You have moved from a 1 in 6 chance of success to a 1 in 3 chance.

Relative Frequency Approach

The relative frequency approach uses the past to make predictions about the future. You look at how many times an event has happened and then look at how many opportunities exist for the event to occur. The formula is the number of times an event occurred divided by the total number of opportunities for the event to occur.

Using Relative Frequency Approach in Business

You can use relative frequency to improve your business decisions. For example if your research shows there are 75 failures for every 100 business startups attempted, you would say that 75 out of 100 startups fail. This reduces to 3/4. That would mean 3 out of 4 startups fail. If you don't do something to change your odds, you can expect that failure probability. This mathematical reality can give you a sense of urgency in your efforts to be the 1 out of 4 that succeeds. In fact, you could study the successes to see how they changed the odds in their favor.

__________________________________________________________________________

Write a summary or highlights from the article. (100 to 120 words)

Find a random variable in your day-to-day life, call it X(ω), and do the following:

• Describe X as either quantitative, qualitative, discrete, continuous, etc.

• Give the support of X (i.e. its possible range of values)

• Speculate on its distribution. Is it normal, geometric, exponential, etc. Give specific reasons and justification for this speculation!

• Sample this random variable at least 5 times.

• Use this sample to estimate its parameters.

• Give the newly parameterized distribution explicitly.

Study results: At study close, the incidence of diabetes was 11.0 cases per 100-person years in the placebo group, 7.8 cases per 100-person years in the metformin group and 4.8 cases per 100-person years in the lifestyle group. The lifestyle intervention reduced the incidence by 58% (95% CI, 48 – 66%), and meftormin by 31% (95% CI, 17-43%), as compared to placebo. P<.001 for each comparison.

f. Were the above results statistically significant?

    How so?

g. Was one intervention more effective than the other?

    Which one?

     Explain.

h. Why was per 100-person-years used as opposed to per 100 persons in this study?

i. Describe how health professionals can implement the results of this study in their practice.

Suppose you are the Chief Marketing Officer for a retailer that has data on the home addresses of its

1,000,000 most active customers. You hope to determine whether sending out “20% off your entire

purchase” coupons by mail will increase revenues.

You conjecture that customers who have access to this coupon will spend more in the store over the

next year. However, skeptics in your company argue that the coupons will just allow customers to

spend less on items they would have purchased anyway. This is a debate that an experiment can

resolve.

In thinking about how large of an experiment you need to have enough statistical power, you realize

that many of the customers you send the coupons to in the mail will not open the mail and so will not

realize they received the coupon.

1. The CEO argues that to estimate the effects of the coupons on revenue, you should compare

the difference in revenues from a) people you sent the coupons and who used them and b)

people you sent the coupons but who did not use them. Write a response to your CEO:

describe the flaw with this plan in language the CEO will understand, and advocate for your

proposed experiment.

2. To avoid the cost of sending out coupons you do not need to, you ask the data science team

to plan an experiment just large enough (with just enough statistical power) to reliably detect

a treatment effect if the true effect on those who open the mail and realize they have the

coupon is a $2 increase in revenue over the next year. The data science team tells you that

an experiment with 100,000 people in the treatment group (leaving the remaining 900,000 in

the control group) will be well-powered to detect an overall difference between the entire

treatment and control groups of $2 in revenue over the next year. To send out the minimum

number of coupons required while still having enough statistical power to detect a $2 effect of

opening the mail, can you send out fewer, the same number of, or more coupons than

100,000?

The average selling price of a smartphone purchased by a random sample of 37 customers was ​$314. Assume the population standard deviation was ​$35.

a. Construct a 95​% confidence interval to estimate the average selling price in the population with this sample.

b. What is the margin of error for this​ interval?

a. The 95​% confidence interval has a lower limit of ​$ and an upper limit of ​$ . ​(Round to the nearest cent as​ needed.)

b. The margin of error is ​$ .​ (Round to the nearest cent as​ needed.)

A random sample of 22 college​ men's basketball games during the last season had an average attendance of 5,166 with a sample standard deviation of 1,755. Complete parts a and b below.

a. Construct a 99​% confidence interval to estimate the average attendance of a college​ men's basketball game during the last season.

The 99​% confidence interval to estimate the average attendance of a college​ men's basketball game during the last season is from a lower limit of to an upper limit of . ​(Round to the nearest whole​ numbers.)

b. What assumptions need to be made about this​ population?

A. The only assumption needed is that the population follows the normal distribution.

B. The only assumption needed is that the population distribution is skewed to one side.

C. The only assumption needed is that the population size is larger than 30.

D. The only assumption needed is that the population follows the​ Student's t-distribution.