Statistics and Probability Answers

The random variable X has probability density function f(x)={ax+bx2 , 0<x<1 . If E(X)=0.6 , find (a)P(X<1/2) and (b)var(x).

A communication channel transmits the digits 0 and 1. However due to static noise, the digit transmitted is incorrectly received with probability 0.2
Suppose we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0
And 11111 instead of 1. If the receiver of the message uses majority decoding , what is the probability that the message will be incorrectly decoded.
( by majority decoding we mean that the message is decoded as 0 if there are
At least 3 zeros in the message received and as 1 otherwise.)

1.Suppose A and B are mutually exclusive events for which P(A)=0.3 AND P(B)=0.5 , What is the probability that (a) either A or B occurs and (b)B occurs but A does not.

Question 8
Plavix is a drug that is given to angioplasty patients to help prevent blood clots. A researcher at McMaster University organized a study that involved 12,562 patients in 482 hospitals in 28 countries. All the patients had acute coronary syndrome, which produces mild heart attacks or unstable angina, chest pain that may precede a heart attack. The patients were divided into two equal groups. Group 1 received daily Plavix pills; group 2 received a placebo. After 1 year, 9.3% of patients on Plavix suffered a stroke or new heart attack or had died of cardiovascular disease, compared with 11.5% of those who took the placebo. Can we infer that Plavix is effective?

Question 7
Credit scorecards are used by financial institutions to help decide to whom loans should be granted (see the Applications in banking: Credit Scorecards summary on page 63). An analysis of the records of a random sample of loans at one bank produced the following results:
Score below 600 Score 600 or More
Sample size 562 804
Number defaulted 11 7
Do these results allow us to conclude that those who score below 600 are more likely to default than those who score 600 or more? Use a 10% significance level.

Question 6
A firm has classified its customers in two ways: (1) according to whether the account is overdue and (2) whether the account is new (less than 12 months) or old. To acquire information about which customers are paying on time and which are overdue, a random sample of 292 customer accounts was drawn. Each was categorized as either a new account or an old account, and whether the customer has paid or is overdue. The results are summarized next.
New account Old account
Sample size 83 209
Overdue account 12 49
Is there enough evidence at the 5% significance level to infer that new and old accounts are different with respect to overdue accounts?

Question 5
Many stores sell extended warranties for products they sell. These are very lucrative for store owners. To learn more about who buys these warranties, a random sample was drawn of a store’s customers who recently purchased a product for which an extended warranty was available. Among other variables, each respondent reported whether he or she paid the regular price or a sale price and whether he or she purchased an extended warranty.

Regular Price Sale Price
Sample size 229 178
No who bought extended warranty 47 25
Can we conclude at the 10% significance level that those who paid the regular price are more likely to buy an extended warranty?

Question 4
Who spends more on their vacations, golfers or skiers? To help answer this question, a travel agency surveyed 15 customers who regularly take their spouses on either a skiing or a golfing vacation. The amounts spent on vacations last year are shown here. Can we infer that golfers and skiers differ in their vacation expenses?

Golfer 2450 3860 4528 1944 3166 3275 4490 3685 2950
Skier 3805 3725 2990 4357 5550 4130

A number of restaurants feature a device that allows credit card users to swipe their cards at the table. It allows the user to specify a percentage or a dollar amount to leave as a tip. In an experiment to see how it works, a random sample of credit card users was drawn. Some paid the usual way, and some used the new device. The percent left as a tip was recorded and listed below. Can we infer that users of the device leave larger tips?

Usual: 10.3 15.2 13.0 9.9 12.1 13.4 12.2 14.9 13.2 12.0
Device: 13.6 15.7 12.9 13.2 12.9 13.4 12.1 13.9 15.7 15.4 17.4

A men’s softball league is experimenting with a yellow baseball that is easier to see during night games. One way to judge the effectiveness is to count the number of errors. In a preliminary experiment, the yellow baseball was used in 10 games and the traditional white baseball was used in another 10 games. The number of errors in each game was recorded and is listed here. Can we infer that there are fewer errors on average when the yellow ball is used?

Yellow: 5 2 6 7 2 5 3 8 4 9
White: 7 6 8 5 9 11 8 3 6 10