(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.

Use R to generate n = 400 samples (idependent identically distributed random numbers) of X ∼ N(0, 4). For each Xi , simulate Yi according to Yi = 3 + 2.5Xi + εi , where εi ∼ N(0, 16), i = 1, ..., n. Use R to solve the following questions. (a) Compute the least square estimators of βˆ 0 and βˆ 1. (b) Draw a regression line according to the numbers computed in (a). Plot Y and X with this regression line. Hint: Use rnorm.

A coin with probability p>0 of turning up heads is tossed 4 times. Let X be the number of times heads are tossed.

(a) Find the probability function of X in terms of p.

(b) The result above can be extended to the case of n independent tosses (that is, for a generic number of tosses), and the probability function in this case receives a very specific name. Find the name of this particular probability function.

Notice that the probability of turning up tails in one toss is 1−p.

a)  A coin is flipped 6 times, find the probability of getting exactly 4 heads.  Hint: The Binomial Distribution Table can be very helpful on questions 19-21.  If you use the table for this question, give your answer exactly as it appears.  If you calculated your answer, round to the thousandths place.

b) A coin is flipped 6 times. Find the probability of getting at least 3 heads. If you used a table to help find your answer, give it to the thousandths place. If you used a formula to calculate your answer, round to the thousandths place.

c) A coin is flipped 6 times, find the probability of getting at most 2 heads. If you used a table to help find your answer, give it to the thousandths place. If you used a formula to calculate your answer, round to the thousandths place.

Suppose a road is flooded with probability p = 0.10 during a year and not more than one flood occurs during a year.

What is the probability that it will be flooded at most twice during a 10-year period?

After a group of (volunteer) experimental subjects achieved a specified blood alcohol level from consuming liquor, psychologists placed each subject in a room and threatened them with electric shocks. Using special equipment, the psychologists recorded the startle response time (in milliseconds) of each subject. The mean and standard deviation of the startle response times is 37.9 milliseconds and 12.4 milliseconds, respectively.

a. Without knowing the shape of the distribution of these response times, what is the interval that contains at least 75% of response times?

b. If you are told that the shape of the distribution of response times is normal,

i. What interval contains approximately 68% of startle response times?

ii. What interval contains approximately 95% of startle response times?

iii. What interval contains 99.7% (i.e., almost all) of startle response times?

iv. Approximately what percentage of startle response times fall below 25.5 milliseconds? HINT: Use the fact that a normal distribution is symmetric around the mean to answer this question.