Confidence Intervals

Confidence by flickr user gerriet

A ferry boat captain knows from past experience that 35% of the passengers will get sick in the rough water ahead. The ferry has 126 passengers. What is the probability that at least 50 passengers will get sick?

(a) Solve this as a binomial distribution problem.

A ferry boat captain knows from past experience that 35% of the passengers will get sick in the rough water ahead. The ferry has 126 passengers. What is the probability that at least 50 passengers will get sick?(b) Solve this as a normal distribution problem. i.e. a normal approximation of the binomial distribution.

(c) Compare your results in (a) and (b). How do they compare?

A handy little applet ...

http://onlinestatbook.com/stat_sim/index.html

Solve a Binomial Problem as an Approximation to a Normal Distribution Problem

According to a group promoting safer driving habits, 63% of all drivers in Manitoba wear seatbelts when driving. During a Safety Week road check, 85 cars were stopped. What is the probability that from 50 to 60 (inclusive) drivers were wearing seatbelts.

OLD HOMEWORK (get it done!)

A toy manufacturer produces balloons that have a 3 percent defective rate. In a shipment of 4000 balloons, what is the probability that:

(b) between 100 and 130 balloons inclusive will be defective?

(a) fewer than 100 balloons will be defective?

Use a normal approximation to solve this problem.OLD HOMEWORK (get it done!)

The manager of the Jean Shop knows that 3 percent of all jeans sold will be defective, and the money paid for these pairs of jeans will be refunded. The manager went on holidays for a period of time, and an employee sold 247 pairs of jeans. The employee reported that refunds were given for 14 pairs of jeans.

(c) Does the employer have proof that the employee did something wrong?

(b) Does the employer have reason to be suspicious of the employee?

(a) What is the probability that 14 pairs of jeans were defective?

OLD HOMEWORK (get it done!)

A producer of hatching eggs acknowledges that 4 percent of all the eggs produced will not hatch. In a shipment of 600 eggs, what is the probability that:

(c) between 20 and 24 inclusive will not hatch?

(b) fewer than 20 will not hatch?

(a) at least 25 will not hatch?

OLD HOMEWORK (get it done!)

Mike wants to buy a new electronic machine that can modify CDs and DVDs. The machine has been advertised on TV, is apparently available in the U.S. and parts of Western Europe, but is currently not available in the local stores. The shop manager has ordered some of the machines, but has received no response from the supplier. Mike needs the machine for his business, and so he asks the salesperson when he will get the machine if he orders it now. The table shows a record of the conversation between Mike and the salesperson.

CONFIDENCE INTERVALS

confidence

A Confidence Interval Some Senior 4 students in a large high school want to change a tradition at graduation. Instead of wearing the usual cap and gown, they want to wear formal clothes. A quick survey of 40 randomly selected students shows that 17 prefer formal wear.

In the above example, the sample was fairly small, and almost half of the students preferred formal wear. The students would not likely make a decision based on the results of this survey. They would probably ask some of the following questions:

• How certain can we be that the results would be approximately the same if another survey of 40 students were done?

• Is it possible that a majority of students prefer formal wear?

• Based on the results of this survey, what are the smallest and largest numbers of students that would likely be in favour of dressing formally?

We will study ways of measuring your confidence that an answer is correct, and also the margin of error in the results of a survey.

A confidence interval is the range of the estimate we are making. That is, the confidence interval shows a range of data that includes a certain percentage of the data. For this question, a confidence interval will show the number of students in the sample that prefer formal wear. A 95 percent confidence interval indicates that we are 95 percent certain that the number of students who prefer formal wear is included in the interval.

Some notes about confidence intervals:

• You should be aware that there are several different kinds of confidence intervals. In this lesson, you will study only the above type of confidence interval. Other more complex confidence intervals are beyond the scope of this course.

• We will study a 95 percent confidence interval in this lesson, because this is the level of confidence stated most frequently in the media. A 95 percent confidence interval is the same as data that are considered ". . . accurate 19 times out of 20 . . .".

• A variety of confidence intervals may be constructed using technology, but you will use algebra and technology to determine confidence intervals

Between what two z-scores will you find exactly 95% of all data in a standard normal distribution?

In a normal distribution, what percent of all the data lie within 2 standard deviations of the mean?

The student council is planning a spring dance. They need to sell 85 tickets in order to cover their costs. Records show that, on average, 10% of the students enrolled at the school attend dances. This year there are 1160 students enrolled at the school.Construct a 95% confidence interval for the number of students who will attend the spring dance.

It is known from past experience that, on average, 4.5 percent of all mouse traps produced by a company are defective. If 50 traps produced by the company are selected at random, find the probability that from 3 to 6 inclusive of them are defective.

HOMEWORK

In a small community, 65 percent of the people speak at least two languages. What is the probability that, if a group of 40 people is randomly selected, at most seven of them speak only one language?

HOMEWORK

It is known from past experience that, on average, 4.5 percent of all mouse traps produced by a company are defective. If 50 traps produced by the company are selected at random, find the probability that from three to six inclusive of them are defective.

Use a normal approximation to solve this problem.HOMEWORK