Chapter 7 hand-in Homework
1) Mention three thing about the standard error of the mean

2) Why is the Central Limit Theorem important in statistics?

3) The diameter of a brand of tennis balls is approximately normally distributed, with a mean of 2.63 inches and a standard deviation of 0.03 inch. If you select a random sample of nine tennis balls, 
a) What is the sampling distribution of the mean?   

b) Assume the diameter of a brand of tennis balls was known to have a right skewed distribution with a mean of 2.63 inches and a standard deviation of 0.03 inch. Suppose we collect a random sample of 100 tennis balls. Describe the sampling distribution of the mean diameter for these 100 tennis balls.

c) How will the mean of the sampling distribution compare to the population mean in this problem?

d) How will the standard deviation of the sampling distribution (standard error of the mean) compare to the population standard deviation?

e) Find the approximate probability that the mean diameter of nine tennis balls exceeded 2.61.

f) Find the approximate probability that the mean diameter of the nine tennis balls exceeded 2.68.

g) Find the approximate probability that the mean diameter of the nine tennis balls was no more than 2.60 inches.

4) n a random sample of 64 people, 48 are classified as “successful”.
a) Determine the sample proportion, p, of “successful” people.

b) Determine the standard error of the proportion.

c) What proportion of the samples will have between 20% and 30% of people who will considered “successful”?

d) What proportion of the samples will have less than 75% of people who will be considered “successful”?

e) 90% of the samples will have less than what percentage of people who will be considered “successful”?

f) 90% of the samples will have more than what percentage of people who will be considered “successful”?

A team of visiting polio eradication workers were informed during their orientation session that population-wide studies done in their host country showed that the risk of polio in villages of that country was strongly epidemiologically associated with the village’s economic/human development circumstances, which ranged greatly from village to village.  In some villages, residents lived in hand-constructed huts with no running water, no latrines or sewage disposal areas, and no electricity.  In other places, residents lived in wooden or adobe homes which, though modest by Western standards, had all of the above services in place and whose street side craft shops and food markets did a brisk business, catering both to locals and visitors.

Knowing this information, the team went into several villages and attempted to assign a “human development rating” to each family. This was based on that family’s income situation, access to running water, access to elementary school for their children, and the condition of the home. To their surprise, they found that families in all the villages had no difference in polio risk based on the family’s human development rating.

The following table shows a portion of the monthly returns data (in percent) for 2010–2016 for two of Vanguard’s mutual funds: the Vanguard Energy Fund and the Vanguard Healthcare Fund. [You may find it useful to reference the t table.]

H₀: ρ_xy = 0; H_A: ρ_xy ≠ 0b. Specify the competing hypotheses in order to determine whether the population correlation coefficient is different from zero.

H₀: ρ_xy ≤ 0; H_A: ρ_xy > 0

H₀: ρ_xy ≥ 0; H_A: ρ_xy < 0

c-1. Calculate the value of the test statistic. (Round intermediate calculations to at least 4 decimal places and final answer to 2 decimal places.)

c-2. Find the p-value.

p-value < 0.01

0.01 ≤ p-value < 0.02

0.02 ≤ p-value < 0.05

0.05 ≤ p-value < 0.10

p-value ≥ 0.10

MORE BENEFITS OF EATING ORGANIC

Using specific data, we find a significant difference in the proportion of fruit flies surviving after 13 days between those eating organic potatoes and those eating conventional (not organic) potatoes. This exercise asks you to conduct a hypothesis test using additional data. In this case, we are testing

      H0⁢ : po⁢= pcHa⁢: po⁢ > pc

where po and pc represent the proportion of fruit flies alive at the end of the given time frame of those eating organic food and those eating conventional food, respectively. Use a 5% significance level.

Effect of Organic Potatoes After 11 Days

After 11 days, the proportion of fruit flies eating organic potatoes still alive is 0.69 , while the proportion still alive eating conventional potatoes is 0.66 . The standard error for the difference in proportions is 0.031 .

What is the value of the test statistic?

Round your answer to two decimal places.

 z= 

What is the p-value?

Round your answer to three decimal places.

p-value = Enter your answer; p-value

What is the conclusion?

Choose the answer from the menu;                                                          Reject H_0.Do not reject H_0.

Is there evidence of a difference?

Choose the answer from the menu; Is there evidence of a difference?                                                          YesNo

A physician wants to develop criteria for determining whether a​ patient's pulse rate is​ atypical, and she wants to determine whether there are significant differences between males and females. Use the sample pulse rates below.
Male
96
84
68
76
68
60
60
60
84
88

Female
104
88
72
84
84
72
64
60
76
124

Assume that females have pulse rates that are normally distributed with a mean of mu equals 73.0 beats per minute and a standard deviation of sigma equals 12.5 beats per minute. Complete parts​ (a) through​ (c) below. a. If 1 adult female is randomly​ selected, find the probability that her pulse rate is less than 79 beats per minute. The probability is ___. ​(Round to four decimal places as​ needed.) b. If 4 adult females are randomly​ selected, find the probability that they have pulse rates with a mean less than 79 beats per minute. The probability is ___ ​(Round to four decimal places as​ needed.)

c. Why can the normal distribution be used in part​ (b), even though the sample size does not exceed​ 30?

A. Since the original population has a normal​ distribution, the distribution of sample means is a normal distribution for any sample size.

B. Since the distribution is of​ individuals, not sample​ means, the distribution is a normal distribution for any sample size.

C. Since the mean pulse rate exceeds​ 30, the distribution of sample means is a normal distribution for any sample size.

D. Since the distribution is of sample​ means, not​ individuals, the distribution is a normal distribution for any sample size.

In a study of the domestic market share of the three major automobile manufacturers A, B, and C in a certain country, it was found that their current market shares were 60%, 20%, and 20%, respectively. Furthermore, it was found that of the customers who bought a car manufactured by A, 75% would again buy a car manufactured by A, 15% would buy a car manufactured by B, and 10% would buy a car manufactured by C. Of the customers who bought a car manufactured by B, 90% would again buy a car manufactured by B, whereas 5% each would buy cars manufactured by A and C. Finally, of the customers who bought a car manufactured by C, 85% would again buy a car manufactured by C, 5% would buy a car manufactured by A, and 10% would buy a car manufactured by B. Assuming that these sentiments reflect the buying habits of customers in the future, determine the market share that will be held by each manufacturer after the next two model years. (Round your answers to the nearest percent.)

PLEASE ANSWER ALL THE QUESTIONS!!!!!! DO NOT ONLY ANSWER PART OF THE QUESTION.

ALSO TRY TO ANSWER THE QUESTIONS AS SPECIFIC AS POSSIBLE. SHOW YOUR WORK!!!!!

A sample of 24 recently sold. The variables are: the sale price in $/10000 (Y), the size of the home in sq. ft./1000 (X₁), and the number of rooms (X₂).

a) Calculate SSR(X₂| X₁).

b) Calculate SSR(X₁²| X₁)

c) Test to see if the quadratic term is useful in the model

Y= B0 +B1X1 + B2X1^2 + E

d)   Calculate SSR(X₂, X₁X₂| X₁) and MSR(X₂, X₁X₂| X₁).

e)   Perform a nested models test to test H₀: B2=B3=0 in the model:

Y = B0 + B1X1 + B2X2 +B3X1X2 + E

f)    Which model would you use when trying to predict the sale price?

The following data give the ages (in years) of all six members of a family 8 10 12 50 55 69

a. List all the possible sample of size two (without replacement) that can be selected from this population. Calculate the mean for each of these samples. Write the sampling distribution of sample mean.

b. Compare the mean of a population probability distribution with that of a sampling distribution.

c. Compare the dispersion in the population with that of the sample means.

A nutrition expert claims that the average American is overweight. To test his claim, a random sample of 26 Americans was selected, and the difference between each person's actual weight and idea weight was calculated. For this data, we have x¯=16.1x¯=16.1 and s=30s=30. Is there sufficient evidence to conclude that the expert's claim is true? Carry out a hypothesis test at a 6% significance level.

A. The value of the standardized test statistic:

Note: For the next part, your answer should use interval notation. An answer of the form (−∞,a)(−∞,a) is expressed (-infty, a), an answer of the form (b,∞)(b,∞) is expressed (b, infty), and an answer of the form (−∞,a)∪(b,∞)(−∞,a)∪(b,∞) is expressed (-infty, a)U(b, infty).

B. The rejection region for the standardized test statistic:

C. The p-value is

D. Your decision for the hypothesis test:

A. Do Not Reject H0H0.
B. Reject H0H0.
C. Do Not Reject H1H1.
D. Reject H1H1.

Also, can you add in how I would solve this on a TI-83 calculator? Thanks!

You are trying to predict if audit clients will go bankrupt in the next year. Based on the results from a logistic regression, the actual vs predicted numbers for clients is as follows, when using a 50 percent cutoff probability for predicting that a client will go bankrupt:

One of your colleagues, John, suggests that you should use a 70 percent cutoff probability for predicting that a client will go bankrupt.

Another colleague, Mike, suggests that you should use a 30 percent cutoff probability for predicting that a client will go bankrupt.

Answer the following questions:

(a) Who is correct, in this situation? Explain your answer with appropriate logic.

(b) What will happen to the Actual vs Predicted matrix if you use:

(i) 70 percent cutoff probability suggested by John? That is, how would the total numbers in the two columns and two rows change, if at all they change?

(i) 30 percent cutoff probability suggested by Mike? That is, how would the total numbers in the two columns and two rows change, if at all they change?

2. The dietary intake of vitamin C for adults in New York is not normally distributed, but it has a mean of µ = 88.2 mg/day and standard deviation σ = 12.1 mg/day

(a) What is the probability that a randomly selected New York adult’s vitamin C intake is greater than 100 mg/day? Think carefully about this!

(b) What is the probability that a sample of 100 randomly selected New York adults will have a mean greater than 89 mg/day?

(c) What is the probability that a sample of 200 randomly selected New York adults will have a mean between 88 mg/day and 88.5 mg/day?

The manager of a paint supply store wants to estimate the actual amount of paint contained in 1-gallon cans purchased from a nationally known manufacturer. The manufacturer’s specifications state that the standard deviation of the amount of paint is equal to 0.022 gallon. A random sample of 64 cans is selected, and the sample mean amount of paint per 1-gallon can is 0.988 gallon. a. Construct a 99% confidence interval estimate of the population mean amount of paint included in a 1-gallon can. b. On the basis of your results, do you think that the manager has a right to complain to the manufacturer? Why? c. Must you assume that the population amount of paint per can is normally distributed here? Explain. d. Construct a 95% confidence interval estimate. How does this change your answer to (b)?

1). Write the null and alternative hypothesis. Make sure you write the correct “formula” (H_o or H_a), noting whether it is directional or non-directional depending on whether the question calls for an increase/decrease or simply a change!

2). Tell me what would be a Type I AND a Type II Error for each example. When you explain the type I and type II error for each, make sure you tell me what the conclusion would be and what the reality would be in each scenario.

-You read an article online that suggests that the number of negative tweets posted online about the TV show Game of Thrones significantly increased during the 2019 season, as compared to the previous season. You decide to test this hypothesis.

a) Null H_o:

b) Alternative H_a:

c) Type I Error:

d) Type II Error:

You’re waiting for Caltrain. Suppose that the waiting times have a mean of 12 minutes and a standard deviation of 3 minutes. Use the Chebyshev inequality to answer each of the following questions:

a) What is the largest possible probability that you’ll end up waiting either less than 6 minutes or more than 18 minutes for the train?

b) What is the smallest possible probability that you’ll wait between 6 and 18 minutes for the train?

c) What is the smallest possible probability that you’ll wait between 3 and 21 minutes for the train?

d) What is the smallest possible probability that you’ll wait between 0 and 24 minutes for the train?

e) Based on your answer to part (d), what is the largest possible probability that you’ll need to wait more than 24 minutes for the train? Why is this answer so dramatically different from your answer to #1c above?