Devry MATH399 2019 JULY Week 3 Assignment Introduction to Probability in Statistics

MATH399 Applied Managerial Statistics

Week 3 Assignment Introduction to Probability in Statistics

QuestionYou 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?