Independent random samples of professional football and basketball players gave the following information. Assume that the weight distributions are mound-shaped and symmetric.

Weights (in lb) of pro football players: x₁; n₁ = 21

Weights (in lb) of pro basketball players: x₂; n₂ = 19

(a) Use a calculator with mean and standard deviation keys to calculate x₁, s₁, x₂, and s₂. (Round your answers to one decimal place.)

(b) Let μ₁ be the population mean for x₁ and let μ₂ be the population mean for x₂. Find a 99% confidence interval for μ₁ − μ₂. (Round your answers to one decimal place.)

(c) Examine the confidence interval and explain what it means in the context of this problem. Does the interval consist of numbers that are all positive? all negative? of different signs? At the 99% level of confidence, do professional football players tend to have a higher population mean weight than professional basketball players?

Because the interval contains only negative numbers, we can say that professional football players have a lower mean weight than professional basketball players.Because the interval contains both positive and negative numbers, we cannot say that professional football players have a higher mean weight than professional basketball players.    Because the interval contains only positive numbers, we can say that professional football players have a higher mean weight than professional basketball players.

(d) Which distribution did you use? Why?

The standard normal distribution was used because σ₁ and σ₂ are unknown.

The standard normal distribution was used because σ₁ and σ₂ are known.    

The Student's t-distribution was used because σ₁ and σ₂ are unknown.

The Student's t-distribution was used because σ₁ and σ₂ are known.

What is the probability that if Paul, Mary, and Susan are in a group of 7 people randomly seated in 7 chairs, they want to be in consecutive chairs. What is the probability if the chairs are set in a circle?

A soccer ball manufacturer wants to estimate the mean circumference of​ mini-soccer balls within 0.05 inch. Assume the population of circumferences is normally distributed.

​(a) Determine the minimum sample size required to construct a 95​% confidence interval for the population mean. Assume the population standard deviation is 0.25 inch.

​(b) Repeat part​ (a) using a population standard deviation of 0.35 inch.

​(c) Which standard deviation requires a larger sample​ size? Explain

•Vocabulary list (define)

normal distribution

Gaussian distribution

Standard normal

Z score (or Z value)

•What is the area under a normal distribution?

•What is the area under any distribution?

Find the standard deviation for a set of data that has a mean of 100 and 95% of the data falls between 70 and 130.

** Please show me the procedure, thanks!!!

(1 point) Rework problem 27 from section 3.2 of your text, involving the mice in a cage. For this problem, assume there are 6 grey females, 6 grey males, 5 white females, and 3 white males. As in the book, the biologist selects two mice randomly.

(1) What is the probability of selecting two males given that both are grey?

(2) What is the probability of selecting one male and one female given that both are grey?

Fill in the missing values for this ANOVA summary table round to two decimal places:

he file Utility contains the electricity costs, in dollars, during July of a recent year for a random sample of 50 one-bedroom apartments in a large city: SELF TEST 96 171 202 157 185 90 141 149 206 95 163 150 108 119 183 178 147 116 172 175 123 154 130 151 114 102 153 111 148 128 144 143 187 135 191 197 127 82 213 130 165 168 109 167 166 139 149 137 129 158 Decide whether the data appear to be approximately normally distributed by a. comparing data characteristics to theoretical properties. b. constructing a normal probability plot.

Let X1,X2,...,Xn be a random sample from any distribution with mean μ and moment generating function M(t). Assume that M(t) is finite for some t > 0.

Let c>μ be any constant. Let Yn = X1+X2+···+Xn. Show that P(Yn ≥ cn) ≤ exp[−n a(c)] where P(Yn ≥ cn) ≤ exp[−n a(c)]

a(c) = sup[ct − ln M (t)]. t > 0

The size of the left upper chamber of the heart is one measure of cardiovascular health. When the upper left chamber is enlarged, the risk of heart problems is increased. A paper described a study in which the left atrial size was measured for a large number of children age 5 to 15 years. Based on this data, the authors concluded that for healthy children, left atrial diameter was approximately normally distributed with a mean of 26.7 mm and a standard deviation of 4.7 mm.

(a)

Approximately what proportion of healthy children have left atrial diameters less than 24 mm? (Round your answer to four decimal places.)

(b)

Approximately what proportion of healthy children have left atrial diameters greater than 32 mm? (Round your answer to four decimal places.)

(c)

Approximately what proportion of healthy children have left atrial diameters between 25 and 30 mm? (Round your answer to four decimal places.)

(d)

For healthy children, what is the value for which only about 20% have a larger left atrial diameter? (Round your answer to two decimal places.)

mm

Consider babies born in the "normal" range of 37–43 weeks gestational age. A paper suggests that a normal distribution with mean

μ = 3500 grams

and standard deviation

σ = 710 grams

is a reasonable model for the probability distribution of the continuous numerical variable

x = birth weight

of a randomly selected full-term baby.

(a)

What is the probability that the birth weight of a randomly selected full-term baby exceeds 4000 g? (Round your answer to four decimal places.)

(b)

What is the probability that the birth weight of a randomly selected full-term baby is between 3000 and 4000 g? (Round your answer to four decimal places.)

(c)

What is the probability that the birth weight of a randomly selected full-term baby is either less than 2000 g or greater than 5000 g? (Round your answer to four decimal places.)

(d)

What is the probability that the birth weight of a randomly selected full-term baby exceeds 7 pounds? (Hint: 1 lb = 453.59 g. Round your answer to four decimal places.)

(e)

How would you characterize the most extreme 0.1% of all full-term baby birth weights? (Round your answers to the nearest whole number.)

The most extreme 0.1% of birth weights consist of those greater than  grams and those less than  grams.

(f)

If x is a random variable with a normal distribution and a is a numerical constant

(a ≠ 0),

then

y = ax

also has a normal distribution. Use this formula to determine the distribution of full-term baby birth weight expressed in pounds (shape, mean, and standard deviation), and then recalculate the probability from part (d). (Round your answer to four decimal places.)

How does this compare to your previous answer?

The value is much smaller than the probability calculated in part (d).The value is about the same as the probability calculated in part (d).    The value is much larger than the probability calculated in part (d).

A particular manufacturing design requires a shaft with a diameter of 17.000 ​mm, but shafts with diameters between 16.988 mm and 17.012 mm are acceptable. The manufacturing process yields shafts with diameters normally​ distributed, with a mean of 17.004 mm and a standard deviation of 0.004 mm.

Complete parts​ (a) through​ (d) below.

a. For this​ process, what is the proportion of shafts with a diameter between 16.988mm and 17.000 mm​?

The proportion of shafts with diameter between 16.988 mm and 17.000 mm is 0.1587.

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

b. For this​ process, what is the probability that a shaft is​ acceptable?

The probability that a shaft is acceptable is 0.9772.

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

c. For this​ process, what is the diameter that will be exceeded by only 2.5​% of the​ shafts?

The diameter that will be exceeded by only 2.5​% of the shafts is 17.0118 mm.

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

d. What would be your answers to parts​ (a) through​ (c) if the standard deviation of the shaft diameters were 0.003 ​mm? If the standard deviation is 0.003​mm, the proportion of shafts with diameter between 16.988 mm and 17.000 mm is

Q1: A study of environmental air quality measured suspend particular matter in air samples at two sites. DATA is listed in the table. Site 1 22 68 36 32 42 24 28 38 40 Site 2 38 34 36 40 39 34 33 32 37 (a) Calculate the mean and standard deviation for each group. (5) (b) Test the hypotheses that air quality for two sites are different. (5) Level of Significance= 0.05

1. For a data set with 3 variables and 3 observations, suppose Xbar, the sample mean vector is [5, 3, 4]’. Let b’ = (1 1 1) and c’ = (1 2 -3).

The sample covariance matrix is given as, S = ( 13 −3.5 1.5;  −3.5 1 −1.5 ; 1.5 −1.5 3 )

(a) Find the sample mean and variance for b’X and c’X.

(b) Find the sample mean and variance for c’X.

(c) Find the covariance between b’X and c’X.