We are creating a new card game with a new deck. Unlike the normal deck that has 13 ranks (Ace through King) and 4 Suits (hearts, diamonds, spades, and clubs), our deck will be made up of the following.

Each card will have:
i) One rank from 1 to 15.
ii) One of 5 different suits.

Hence, there are 75 cards in the deck with 15 ranks for each of the 5 different suits, and none of the cards will be face cards! So, a card rank 11 would just have an 11 on it. Hence, there is no discussion of "royal" anything since there won't be any cards that are "royalty" like King or Queen, and no face cards!

The game is played by dealing each player 5 cards from the deck. Our goal is to determine which hands would beat other hands using probability. Obviously the hands that are harder to get (i.e. are more rare) should beat hands that are easier to get.a)

b)How many different ways are there to get exactly 1 pair (i.e. 2 cards with the same rank)?

What is the probability of being dealt exactly 1 pair?
Round your answer to 7 decimal places.

c) How many different ways are there to get exactly 2 pair (i.e. 2 different sets of 2 cards with the same rank)?

What is the probability of being dealt exactly 2 pair?
Round your answer to 7 decimal places.

The Aluminum Association reports that the average American uses 56.8 pounds of aluminum in a year. A random sample of 49 households is monitored for one year to determine aluminum usage. If the population standard deviation of annual usage is 12.1 pounds, what is the probability that the sample mean will be each of the following?

Appendix A Statistical Tables
a. More than 61 pounds
b. More than 57 pounds
c. Between 55 and 58 pounds
d. Less than 55 pounds
e. Less than 48 pound

The U.S. Bureau of Labor Statistics released hourly wage figures for various countries for workers in the manufacturing sector. The hourly wage was $30.67 for Switzerland, $20.20 for Japan, and $23.82 for the U.S. Assume that in all three countries, the standard deviation of hourly labor rates is $3.00.

Appendix A Statistical Tables

a. Suppose 41 manufacturing workers are selected randomly from across Switzerland and asked what their hourly wage is. What is the probability that the sample average will be between $30.00 and $31.00?
b. Suppose 32 manufacturing workers are selected randomly from across Japan. What is the probability that the sample average will exceed $21.00?
c. Suppose 50 manufacturing workers are selected randomly from across the United States. What is the probability that the sample average will be less than $23.00?

(Round the values of z to 2 decimal places. Round your answers to 4 decimal places.)

1. explain the difference between creating a sampling distribution with n > 1 vs. plotting individual data points.

2.  explain why it makes sense that increasing the sample size should decrease the standard deviation in a normal distribution. Explain what this does to the overall shape of the normal curve

How do you find a stanine with normal distribution? Can you provide an example.

(A) If the answer in (a) is greater than the answer in (c) then we cannot conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average.

(B) If the answer in (b) is greater than the answer in (c) then we conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average.

(C) If the answer in (a) is greater than the answer in (b) then we cannot conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average

(D) If the answer in (c) is greater than 0.04 then we conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average

(E) If the answer in (c) is less than 0.04 then we cannot conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average

(F) If the answer in (c) is less than 0.04 then we conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average.

(G) If the answer in (b) is greater than the answer in (c) then we cannot conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average

(H) If the answer in (a) is greater than the answer in (c) then we conclude at the 4% significance
level that the average birthweight from this hospital is different from the national average.

2. Calculate the 99%, 95%, and 90% confidence interval for the following information. Identify how these confidence intervals are similar and how they are different. Explain why. (70 points)

x̄ = 55 s = 15 n = 101

1. The 99% Confidence Interval:

2. The 95% Confidence Interval:

3. The 90% Confidence Interval:

4. Similarities:

5. Differences:

6. Why?

According to a study published in the New England Journal of Medicine, overweight people on low-carbohydrate and Mediterranean diets lost more weight and got greater cardiovascular benefits than people on a conventional low-fat diet (The Boston Globe, July 17, 2008). A nutritionist wishes to verify these results and documents the weight loss (in pounds) of 30 dieters on the low-carbohydrate and Mediterranean diets and 30 dieters on the low-fat diet. Let Low-carb or Mediterranean and Low-fat diets represent populations 1 and 2, respectively.

a. Set up the hypotheses to test the claim that the mean weight loss for those on low-carbohydrate or Mediterranean diets is greater than the mean weight loss for those on a conventional low-fat diet.

• H₀: μ₁ − μ₂ = 0; H_A: μ₁ − μ₂ ≠ 0

• H₀: μ₁ − μ₂ ≥ 0; H_A: μ₁ − μ₂ < 0

• H₀: μ₁ − μ₂ ≤ 0; H_A: μ₁ − μ₂ > 0

b-1. Calculate the value of the test statistic. Assume that the population variances are unknown but equal. (Round your answer to 3 decimal places.)

b-2. Find the p-value

• p-value < 0.01

• 0.01 ≤ p-value < 0.025

• 0.025 ≤ p-value < 0.05

• 0.05 ≤ p-value < 0.10

• p-value ≥ 0.10

c. At the 5% significance level, can the nutritionist conclude that people on low-carbohydrate or Mediterranean diets lose more weight, on average, than people on a conventional low-fat diet?

• Yes

• No

Assume that the Poisson process X = {X(t) : t ≥ 0} describes students’ arrivals at the library with intensity λ = 4 per hour. Given that the tenth student arrived exactly at the end of fourth hour, or W₁₀ = 4, find:

1. E [W₁|W₁₀ = 4]

2. E [W₉ − W₁|W₁₀ = 4].

Hint: Suppose that X {X(t) : t ≥ 0} is a Poisson process with rate λ > 0 and its arrival times are defined for any natural k as W_k = min[t ≥ 0 : X(t) = k] (1) Then for any natural m, the inter-arrival times, {T₁ = W₁, T₂ = W₂ − W₁, . . . , T_m = W_m − W_m−1} are independent variables with the common exponential distribution, f^T(t) = λ · e −λ·t for t > 0.

Two plots at Rothamsted Experimental Station were studied for production of wheat straw. For a random sample of years, the annual wheat straw production (in pounds) from one plot was as follows.

Use a calculator to verify that, for this plot, the sample variance is s² ≈ 0.274.

Another random sample of years for a second plot gave the following annual wheat production (in pounds).

Use a calculator to verify that the sample variance for this plot is s² ≈ 0.761.

Test the claim that there is a difference (either way) in the population variance of wheat straw production for these two plots. Use a 5% level of signifcance.

(a) What is the level of significance?

State the null and alternate hypotheses.

H_o: σ₁² = σ₂²; H₁: σ₁² > σ₂²H_o: σ₁² > σ₂²; H₁: σ₁² = σ₂²    H_o: σ₂² = σ₁²; H₁: σ₂² > σ₁²H_o: σ₁² = σ₂²; H₁: σ₁² ≠ σ₂²

(b) Find the value of the sample F statistic. (Use 2 decimal places.)


What are the degrees of freedom?

What assumptions are you making about the original distribution?

The populations follow independent normal distributions.The populations follow independent chi-square distributions. We have random samples from each population.    The populations follow independent normal distributions. We have random samples from each population.The populations follow dependent normal distributions. We have random samples from each population.

(c) Find or estimate the P-value of the sample test statistic. (Use 4 decimal places.)

p-value > 0.2000.100 < p-value < 0.200    0.050 < p-value < 0.1000.020 < p-value < 0.0500.002 < p-value < 0.020p-value < 0.002

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

At the α = 0.05 level, we reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.05 level, we reject the null hypothesis and conclude the data are statistically significant.    At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are not statistically significant.At the α = 0.05 level, we fail to reject the null hypothesis and conclude the data are statistically significant.

(e) Interpret your conclusion in the context of the application.

Fail to reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots.Reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.    Reject the null hypothesis, there is sufficient evidence that the variance in annual wheat production differs between the two plots.Fail to reject the null hypothesis, there is insufficient evidence that the variance in annual wheat production differs between the two plots.

1. The time between arrivals of oil tankers at a loading dock at Prudhoe Bay is given by the following probability distribution:

Time Between Ship Arrivals (days)          Probability

1                                                                                    0.05

2                                                                                    0.10

3                                                                                    0.15

4                                                                                    0.25

5                                                                                    0.25

6                                                                                    0.15

7                                                                                    0.05

1.00

The time required to fill a tanker with oil and prepare it for sea is given by the following probability distribution:

Time to Fill and Prepare (days)                Probability

2                                                                                    0.10

3                                                                                    0.30

4                                                                                    0.40

5                                                                                    0.20

1.00

1. Simulate the movement of tankers to and from the single loading dock for the first 100 arrivals. Compute the average time between arrivals, average waiting time to load, and average number of tankers waiting to be loaded. (hint: The COUNTIF function to count # of tankers in the system upon the arrival of the 11th tanker, =COUNTIF(A1:A10,">"&B11) // count cells A1:A10 greater than value in B11

A1:A10 = the departure times of tankers 1:10

B11 = the arrival time of tanker 11

2. Discuss any hesitation you might have about using your results for decision making.

​(a)

chiχsquared2equals=nothing

​(Round to two decimal places as​ needed.)

Vequals=nothing

​(Round to two decimal places as​ needed.)

​(b) Interpret the values from part​ (a). What do they indicate about the​ association?

Since

▼

Upper VV

chi squaredχ2

is

▼

somewhat large

rather small

large

​, there is

▼

strong

weak

moderate

association between the two variables.

chart with given information

Premium - Weekday 127 Weekend 124 Total 251

Plus- Weekday 100 Weekend 203 Total 303

Regular-Weekday 487 Weekend 128 Total 615

Total-Weekday 714 Weekend 455 Total 1169

Since V or X to the second is somewhat large rather small or large there is strong weak or moderate association between the two variables.

An investigator in the Statistics Department of a large university is interested in the effect of exercise in maintaining mental ability. She decides to study the faculty members aged 40 to 50 at his university, looking separately at two groups: The ones that exercise regularly, and the ones that don’t. There turn out to be several hundred people in each group, so she takes simple random sample of 25 persons from each group, for detailed study. One of the things she does is to administer an IQ test to the sample people, with the following results: Regular Exercise No Regular Exercise Sample size 25 25 Average score 130 120 Standard deviation 15 15 The investigator concludes that exercise does indeed help to maintain mental ability among the faculty members aged 40 to 50 at his university. Is this conclusion justified? Explain whether you agree with her and show your reasoning mathematically. (20 points)

Seventy homes that were for sale in Gainesville, Florida in Spring of 2019 were randomly selected. A regression model to predict house price in thousands was run based on first floor square footage and the indicator variable for NorthWest (1 if the house was in the NW, O if not).

Find the predicted house price in thousands for a house with 1,953 square feet on the first floor and in the SW. Round your answer to two decimal points.