1. The number of groups visiting a remote hut in the woods is given by a random variable X with the following probability dis- tribution (asume there are never more than 4 groups visiting) 01 2 3 P(X) 350 240 270 (a) What value should replace the question mark to make this a valid distribution (b) Find the probability that at least one group visits the hut in a given day

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1. The number of groups visiting a remote hut in the woods is
given by a random variable X with the following probability dis-
tribution (assume there are never more than 4 groups visiting)
X
P(X)350 240 270?050
(b) Compute and give the two-way table of expected counts un-
der the assumption that the size of gift and donation status
are independent.
2 |3| 4
(c) Use the given table, the table from part b, and your calcu-
lator to carry out a hypothesis test to determine if there is
convincing evidence that the proportions in the two donation
categories are not the same for all three types of 1equesta.
Report the P-value, and state your conclusion with a signif-
icance level of 01.
(a) What value shoukd replace the question mark to make this a
valid distribution?
(b) Find the probability that at least one group visits the hut in
a given day.
4 A machine that cuts corks for wine bottles operates in such a way
that the distribution of the diameter for the corkai is well approx-
imated by a normal distribution with mean 3 cm and standard
deviation (0.2 em. The specifications cal for corks with diameters
between 2.9 and 3.1 cm. What proportion of corks produced by
this machine are defective?
(e) Find the mean uumber of groups that will visit the hut on a
given day.
2. At a large high school, 30% of students play at least one varsity
sport.
(a) What is the expected mumber of students at the high school
you would need to interview before you found one that was
not a varsity athlete.
(b) A random sample of 30 students is taken. Find the proba-
bility that at least 10 are varsity athletes.
(e) A random sample of 30 students is taloen. Find the mean
number of varsity athletes selected and the standard devia-
5. An environmental group the believes that an industrial plant may
be exceeding pollution limits. They take four readings to analyze
whether the true mean discharge of wastewater exceeded the le-
gal limit of 600 gallons per hour. The environmental group uses a
one-sided hypothesis test with a = 05. For each of the following
statements, say whether the statement is true or false and explain
tion.
your answer.
3. A study was done to determine whether including a gift with a
request for a donation affects the proportion who will make a
donation. Suppose the following results were obtained.
(a) If the plant is actually discharging an average of 600 gal-
lons per houns, there is only a 5% chance that the group will
conclude that the plant is exceeding the limit.
(b) If the P-value is P = 03, the environmental group should
conclude that the plant is within legal limits.
Donation No Donation
280
252
No Gift
40
(c) If the plant is exceeding the limit, there is only a 5% chance
that we will conclude that it is exceeding the limit.
Small Gift
46
Large Gift
68
247
(a) Suppose a random participant from the study is selected
What is the probability that they were sent a large gift, given
that they made a donation?
Transcribed Image Text:1. The number of groups visiting a remote hut in the woods is given by a random variable X with the following probability dis- tribution (assume there are never more than 4 groups visiting) X P(X)350 240 270?050 (b) Compute and give the two-way table of expected counts un- der the assumption that the size of gift and donation status are independent. 2 |3| 4 (c) Use the given table, the table from part b, and your calcu- lator to carry out a hypothesis test to determine if there is convincing evidence that the proportions in the two donation categories are not the same for all three types of 1equesta. Report the P-value, and state your conclusion with a signif- icance level of 01. (a) What value shoukd replace the question mark to make this a valid distribution? (b) Find the probability that at least one group visits the hut in a given day. 4 A machine that cuts corks for wine bottles operates in such a way that the distribution of the diameter for the corkai is well approx- imated by a normal distribution with mean 3 cm and standard deviation (0.2 em. The specifications cal for corks with diameters between 2.9 and 3.1 cm. What proportion of corks produced by this machine are defective? (e) Find the mean uumber of groups that will visit the hut on a given day. 2. At a large high school, 30% of students play at least one varsity sport. (a) What is the expected mumber of students at the high school you would need to interview before you found one that was not a varsity athlete. (b) A random sample of 30 students is taken. Find the proba- bility that at least 10 are varsity athletes. (e) A random sample of 30 students is taloen. Find the mean number of varsity athletes selected and the standard devia- 5. An environmental group the believes that an industrial plant may be exceeding pollution limits. They take four readings to analyze whether the true mean discharge of wastewater exceeded the le- gal limit of 600 gallons per hour. The environmental group uses a one-sided hypothesis test with a = 05. For each of the following statements, say whether the statement is true or false and explain tion. your answer. 3. A study was done to determine whether including a gift with a request for a donation affects the proportion who will make a donation. Suppose the following results were obtained. (a) If the plant is actually discharging an average of 600 gal- lons per houns, there is only a 5% chance that the group will conclude that the plant is exceeding the limit. (b) If the P-value is P = 03, the environmental group should conclude that the plant is within legal limits. Donation No Donation 280 252 No Gift 40 (c) If the plant is exceeding the limit, there is only a 5% chance that we will conclude that it is exceeding the limit. Small Gift 46 Large Gift 68 247 (a) Suppose a random participant from the study is selected What is the probability that they were sent a large gift, given that they made a donation?
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