DEVRY MATH399 Week 3 Assignment Introduction to Probability in Statistics Latest 2019 JULY Question # 00603571 Subject: Mathematics Due on: 08/10/2019 Posted On: 08/10/2019 05:34 AM Tutorials: 1 Rating: 4.7/5

MATH399 Applied Managerial Statistics

Week 3 Assignment Introduction to Probability
in Statistics

Question You
toss a coin three times. If you toss heads exactly two times, you win $2. If
you toss heads all three times, you win $8. Otherwise, you lose $3. What is the
expected payout for one round of this game?

• Round your answer to the nearest
cent.

• Enter an expected loss as a
negative number.

QuestionIn
a day care center each child takes three classes and either passes or fails
each class. The grades are shown in the table below. P represents a passed
class, and F represents a failed class. Let Xrepresent the number of classes
failed by a child. Construct a probability distribution for X. Arrange xin
increasing order and write the probabilities P(x) as simplified fractions.

Child Grades

Child 1 PFP

Child 2 PPF

Child 3 FFP

Child 4 PPP

Child 5 PFP

Child 6 PPF

Child 7 FPF

Child 8 PPP

QuestionFor
the probability distribution of X given below, find m.

• Enter m as a decimal rounded to one
decimal place.

x 1 2 3 4

P(x) m 2m 3m 4m

QuestionA
farmer claims that the average mass of an apple grown in his orchard is 100g.
To test this claim, he measures the mass of 150 apples that are grown in his
orchard and determines the average mass per apple to be 98g. The results are
calculated to be statistically significant at the 0.01 level. What is the
correct interpretation of this calculation?

The data
are not statistically significant at the 0.05 level.

The mean mass of any 150 apples grown in the
farmer’s orchard is 98g.

At the 0.01 level of significance, the mean
mass of the apples grown in the farmer’s orchard is different from 100g.

At the 0.01 level of significance, the mean
mass of the apples grown in the farmer’s orchard is 98g.

QuestionA
hospital takes record of any birth that occurs there every day. On one day, the
hospital reports that 35 of the 62 babies born were girls. Assuming that all of
the parents did not have any gender selection procedures, there is a
probability of 0.31 of getting these results by chance. Do these results have
statistical significance at the 0.05 level of significance?

Yes, the
probability of the results occurring is less than 0.05.

No, the probability of the results occurring
is less than 0.05.

Yes, the probability of the results occurring
is greater than 0.05.

No, the probability of the results occurring
is greater than 0.05.

QuestionA
poll was conducted to determine if there was any possible connection between
men and women who live in a certain city and the favorability of a mayor in the
city. In the poll, of the 400 men selected, 210 reported being in favor of the
mayor. Of the 400 women selected, 190 also reported being in favor of the
mayor. Assuming gender has nothing to do with favorability, the probability of
these results occurring by chance is calculated to be about 0.16. Interpret the
results of the calculation at the 0.05 level of significance.

It is
expected that 210 of every 400 men in this city will be in favor of the mayor.

The difference in the proportions is not
statistically significant because the probability is greater than 0.05.

Gender has no relation to the favorability of
this mayor.

The difference in the proportions shows that
gender could have an association with how favorable the mayor is.

QuestionA
city council sends out a survey to city residents to determine the support for
a new school being built downtown. The survey asks residents whether they
support the new school being built and their location in town. Out of those who
respond, 376 of the 641 residents surveyed in Ward 1 support the new school
being built, while 214 of the 398 residents surveyed in Ward 2 support the new
school. Assuming the location of residents has no association with supporting
the new school being built, the probability of the results of the survey being
due to chance is calculated to be 0.12. Interpret the result of this
calculation.

The council
can expect 376 of every 641 residents in Ward 1 to be in favor of building the
new school.

The results of the survey are statistically
significant at the 0.05 level in showing that the location of city residents
has an association with support for building the new school.

The results of the survey are not
statistically significant at the 0.05 level in showing that the location of the
city’s residents has an association with support for building the new school.

The location of city residents is not
associated with support for the new school.

QuestionA
recent survey was conducted to determine if political party has an impact on
whether a person believes that higher education is necessary for career
advancement. In the survey, 410 Republicans and 409 Democrats were surveyed. Of
the Republicans, 36% believed that higher education was necessary for people to
advance their career. This is very different from the 65% of Democrats surveyed
who stated that higher education was necessary. The calculations were
statistically significant at the 0.01 level of significance. What is the
correct interpretation of the probability?

We expect
that 65% of all Democrats believe that higher education is necessary for career
advancement.

At the 0.01 level of significance, political
party is associated with whether a person believes that higher education is
necessary for career advancement.

We cannot say that the results are
statistically significant at the 0.05 level of significance.

At the 0.01 level of significance, political
party determines whether a person believes that higher education is necessary
for career advancement.

Explain the
Role of Probability in Statistics

We see probability used on a regular basis.
The purpose of probability models is to analyze samples (sometimes very large
samples) to make predictions about various types of populations. For example, social media platforms analyze
our interest and place targeted ads based on our use of their platforms.
Meteorologists use various models in an attempt to predict our weather forecasts.
However, have you ever seen an ad and wondered “Why am I seeing
this?”; or, have you ever seen a weather forecast and said “That
forecast was completely wrong!” Probability is a prediction of what we
expect to happen–not a guarantee of what will happen. Oftentimes,
meteorologists look at multiple forecast models and select the one that has the
highest probability of occurring but sometimes a few models have a similar
probability of occurrence. So how do meteorologists make their decision? They
rely on statistical significance. We’ll discuss this in more detail below.

Suppose you want to test whether a die is
fair. Every outcome of a fair die is equally likely to occur, with each
possible outcome having a probability of 16. If you toss a die 120 times, and
of those times, you roll a 2 a total of 16 times, is the die unfair? No. While
we expect to see near 20 of the 120 die rolls result in a 2 in this experiment,
this doesn’t mean that every time we toss a die 120 times we will roll a 2
exactly 20 times. In some experiments, we may roll a 2 more than 20 times, and
in others we may roll it fewer than 20 times.

In another
experiment, if you toss a die 600 times, you should expect to roll a 2 around
100 times. However, let’s say that in one experiment you roll a 2 a total of
160 times. This is very different to the 100 you expected to roll in the
experiment. Although it is possible that 160 rolls of 2 can happen in this
experiment, it is much more likely that the die is weighted in a way that makes
2 more likely to be rolled. If the difference between the results and the
expectation of an experiment is unlikely to be due to chance, the difference is
statistically significant.

Key Terms

• Statistically significant: if
measurements or observations in a statistical experiment are unlikely to have
occurred by chance

Example 1

Question:

The arrest
log for a county newspaper states that 26 of the 42 arrests that occurred in
the past week happened in the same town. Suppose that the population size of
this town is similar to those of other towns in this county. Is this
statistically significant?

Example 2

Question:

A
researcher conducts an experiment to test whether this year’s flu shot is
effective in preventing the flu. He asks the people he surveys whether they received
the flu shot and whether they were diagnosed with the flu during the year. He
surveys 200 people who received the flu shot and 200 people who did not receive
the flu shot. He finds that of the people he surveyed, 12 who received the flu
shot were diagnosed, while 16 who did not receive the shot were diagnosed. From
the survey, can we conclude that the flu shot is effective at preventing the
flu?

Example 3

Question:

A recent
poll was taken in the United States to determine how favorable the president
is. In the state of Florida, 46% of the 500 people polled were in favor of the
president. Meanwhile, in the state of Oregon only 37% of the 500 people polled
were in favor of the president. When calculated, the probability of the
difference in percentage being the result of chance is less than 0.05.
Interpret the result of this calculation.

QuestionA
study was conducted to see if economic class was associated with the highest
completed level of education. According to the study, 155 of 500 lower class
adults obtained a bachelor’s degree or higher. This compares to 167 of 450
middle class adults who obtained a bachelor’s degree or higher. Assuming that
the highest completed level of education does not depend on economic class, the
probability of the data being the result of chance is calculated to be 0.05.
Interpret this calculation.

We can
expect 167 of any 450 middle class adults to have obtained a bachelor’s degree
or higher.

The data is statistically significant in
showing that class is associated with the highest level of education completed.

The data is statistically significant in
showing that class determines the highest level of education completed.

The proportion of lower class adults who
completed bachelor’s degrees or higher is always less than the proportion of
middle class who completed bachelor’s degrees or higher

QuestionYou
bet $50 on 00 in a game of roulette. If the wheel spins 00, you have a net win
of $1,750, otherwise you lose the $50. A standard roulette wheel has 38 slots
numbered 00, 0, 1, 2, … , 36. What is the expected profit for one spin of the
roulette wheel with this bet?

• Round your answer to the nearest
cent.

• Enter an expected loss as a
negative number.

QuestionAccording
to a recent poll, 40.5% of people aged 25 years or older in the state of
Massachusetts have a bachelor’s degree or higher. The poll also reported that
30.0% of people aged 25 years or older in the state of Delaware have a
bachelor’s degree or higher. The poll sampled 354 residents of Massachusetts
and 210 residents of Delaware. The data was calculated to be significant at the
0.013level. Determine the meaning of this significance level.

At the
0.013 level of significance, a larger percentage of residents from
Massachusetts have bachelor’s degrees.

It is not unusual to see 30.0% of a sample of
210 residents of Delaware have bachelor’s degrees because level of education
varies.

It is certain that more residents of
Massachusetts have bachelor’s degrees than do residents of Delaware.

We can expect about 40.5% of any group of 354
Massachusetts residents to have a bachelor’s degree or higher.

QuestionBefore
a college professor gave an exam, he held a review session, where 30 of his 150
students attended the review. The mean score of the students who attended was
86%, whereas the mean score of the students who didn’t attend the review was
79%. The difference in the mean scores is significant at the 0.05 level,
assuming the review session does not associate with a higher exam score.
Determine the meaning of this significance level.

It is not
unusual to see the mean exam score of 120 students be 79% because the testing
abilities of students vary.

We expect the mean score of a group of 30
students who attend a review session to be 86%.

At the 0.05 level of significance, attendance
of the review session is associated with a higher exam score.

The review session is helpful to students at
the 0.01 level of significance.

QuestionA
health survey determined the mean weight of a sample of 762 men between the
ages of 26 and 31to be 173 pounds, while the mean weight of a sample of 1,561
men between the ages of 67 and 72was 162 pounds. The difference between the
mean weights is significant at the 0.05 level. Determine the meaning of this
significance level.

We expect
that the mean weight of any sample of 762 men between the ages of 26and 31 is
173 pounds.

It is not unusual to see the mean weight of
1,561 men between the ages of 67 and 72 to be 162 pounds because the weight of
men varies.

The results are statistically significant at
the 0.01 level.

At the 0.05 level of significance, the age of
men has an association with their mean weight.

QuestionA
hat contains 6 red balls, 4 yellow balls, and 2 green balls. If you draw a red
ball, you lose $5. If you draw a yellow ball, you win $1. If you draw a green
ball, you win $7. What is the expected profit of one draw? Round your answer to
the nearest cent. Enter an expected loss as a negative number.

QuestionA
poll is conducted to determine if political party has any association with
whether a person is for or against a certain bill. In the poll, 214 out of 432
Democrats and 246 out of 421 Republicans are in favor of the bill. Assuming
political party has no association, the probability of these results being by
chance is calculated to be 0.01. Interpret the results of the calculation.

We can
expect that 246 out of every 421 Republicans are in support of this bill.

We cannot say the results are statistically
significant at the 0.05 level.

At the 0.01 level of significance, political
party is associated with whether a person supports this bill.

At the 0.01 level of significance, political
party determines whether a person supports this bill.

QuestionFor
a certain animal, suppose that the number of babies born is independent for
each pregnancy. This animal has a 70% chance of having 1 baby and a 30% chance
of having 2 babies at each pregnancy. Let X be a random variable that
represents the total number of babies if the animal gets pregnant twice.
Construct a table showing the probability distribution of X. Arrange x in
increasing order.

• Write the probabilities P(x) as
decimals rounded to two decimals.

$$2

$$3

$$4

$$0.49

$$0.42

$$0.09

QuestionDetermine
whether or not the distribution is a probability distribution and select the
reason(s) why or why not.

x 2 4 6

P(x) 15 15 15

The given
distribution is not a probability distribution, since the sum of probabilities
is not equal to 1.

The given
distribution is a probability distribution, since the sum of probabilities is
equal to 1.

The given
distribution is not a probability distribution, since at least one of the
probabilities is greater than 1 or less than 0.

The given
distribution is a probability distribution, since the probabilities lie
inclusively between 0 and 1.

QuestionOn
a multiple-choice quiz, a correct answer is awarded 4 points, but an incorrect
answer costs the student 1 point. Suppose each question on the quiz has 4
choices and no question has multiple correct answers. If a student were to
guess on every question, what is the number of points the student should expect
to get per question?