In: Statistics and Probability

4. Let X equal the weight (in grams) of a Hershey’s grape-flavored Jolly Rancher. Denote the median of X by m. We wish to test the null hypothesis H0 : m = 5.900 against the alternative Ha : m > 5.900. A random sample of size n = 25 yielded the following ordered data:

5.625 5.665 5.697 5.837 5.863 5.870 5.878 5.884 5.908 5.967 6.019 6.020 6.029 6.032 6.037 6.045 6.049 6.050 6.079 6.116 6.159 6.186 6.199 6.307 6.387

Explain step by step how to use a t test to test the hypothesis.

Ho : µ = 5.9

Ha : µ > 5.9

sample std dev , s =
0.184680517

Sample Size , n = 25

Sample Mean, x̅ = 6.00

degree of freedom= DF=n-1=
24

Standard Error , SE = s/√n =
0.0369

t-test statistic= (x̅ - µ )/SE = 2.608

α=0.05

critical t value, t* = 1.7109
[Excel formula =t.inv(α,df) ]

(also, p-value=0.0077)

Conclusion: t-stat >t-criitcal value, Reject null hypothesis

Will Smith consumes chocolate and milk. Let y denote chocolate
and x denote milk. Chocolate has an unusual market where there is
only one supplier, and the more chocolate you buy from the
supplier, the higher the price she charges per unit. In fact,
y units of chocolate will cost Will y2
dollars. Milk is sold in the usual way at a price of 2 dollars per
unit. Will’s income is 20 dollars and his utility function is
U =...

Let f(t) be the weight (in grams) of a solid
sitting in a beaker of water after t minutes have elapsed.
Suppose that the solid dissolves in such a way that the rate of
change (in grams/minute) of the weight of the solid at any time
t can be determined from the weight using the
forumula:
f′(t)=−4f(t)(6+f(t))
If there is 1 grams of solid at time t=2 estimate the
amount of solid 1 second later.

Two fair dice are rolled at once. Let x denote the difference in
the number of dots that appear on the top faces of the two dice.
For example, if a 1 and a 5 are rolled, the difference is 5−1=4, so
x=4. If two sixes are rolled, 6−6=0, so x=0. Construct the
probability distribution for x. Arrange x in increasing order and
write the probabilities P(x) as simplified fractions.

Let the random variable X denote the time (hours) for which a
part is waiting for the beginning of the inspection process since
its arrival at the inspection station, and let Y denote the time
(hours) until the inspection process is completed since its arrival
at the inspection station. Since both X and Y measure the time
since the arrival of the part at the inspection station, always X
< Y is true. The joint probability density function for X...

Suppose two fair dice are rolled. Let X denote the product of
the values on the dice and Y denote minimum of the two dice.
Find E[X] and E[Y]
Find Var X and Var Y
Let Z=XY. Find E[Z].
Find Cov(X,Y) and Corr(X,Y)
Find E[X|Y=1] and E[Y|X=1]

SHOW ALL STEPS:
Let X be the live weight of a randomly selected 4-month old male
lamb. Suppose X follows a normal distribution with a mean of 38 kg
and a standard deviation of 2.5 kg. In this question, we consider a
sample of sixteen 4-month old male lambs. Denote by X¯ the mean
live weight of these 16 lambs.
(a) State the distribution, with the corresponding parameters,
of X¯ .
(b) Manually calculate the probability that the mean live...

Let X equal the outcome (1, 2 , 3 or 4) when a fair four-sided
die is rolled; let Y equal the outcome (1, 2, 3, 4, 5 or 6) when a
fair six-sided die is rolled. Let W=X+Y.
a. What is the pdf of W?
b What is E(W)?

Let X denote the distance (m) that an animal moves from its birth site to the first territorial vacancy it encounters. Suppose that for banner-tailed kangaroo rats, X has an exponential distribution with parameter λ = 0.01362. (a) What is the probability that the distance is at most 100 m? (b) What is the probability that distance exceeds the mean distance by more than 2 standard deviations? (c) What is the value of the median distance? (Round your answer to two decimal places.)

Let X denote the amount of space occupied by an article
placed in a 1-ft3 packing container. The pdf of
X is below.
f(x) =
56x6(1 − x)
0
< x < 1
0
otherwise
(a) Graph the pdf.
Obtain the cdf of X.
F(x) =
0
x < 0
0 ≤ x ≤ 1
1
x > 1
Graph the cdf of X.
(b) What is P(X ≤ 0.65) [i.e., F(0.65)]?
(Round your answer to four decimal places.)...

A fair coin is tossed four times. Let X denote the number of
heads occurring and let Y denote the longest string of heads
occurring. (i) determine the joint distribution of X and Y (ii)
Find Cov(X,Y) and ρ(X,Y).

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