A toy manufacturer wants to know how many new toys children buy each year. A sample of 686 children was taken to study their purchasing habits. Construct the 85% confidence interval for the mean number of toys purchased each year if the sample mean was found to be 6.8. Assume that the population standard deviation is 2.1. Round your answers to one decimal place.

A computer software company would like to estimate how long it will take a beginner to become proficient at creating a graph using their new spreadsheet package. Past experience has indicated that the time required for a beginner to become proficient with a particular function of the new software product has an approximately normal distribution with a standard deviation of 21 minutes. Find the sample size necessary to estimate the true average time required for a beginner to become proficient at creating a graph with the new spreadsheet package to within 4 minutes with 90% confidence.

In this section we will turn our attention to comparing two population proportions. Once again, there are times when we aren't necessarily focused on the exact proportion, but rather how proportions from two populations compare, that is, if they are equal, or if one is larger than the other.

When we were comparing population means, we constructed a confidence interval for the difference between the two population means. Similarly, when comparing two population proportions, we use a confidence interval for the difference between the population proportions. The best point estimate for the difference is pˆ1−pˆ2

. In this section we will restrict our discussion to comparing two population proportions when the following conditions are met. Notice that the conditions are similar to those discussed for estimating a single population proportion.

All possible samples of a given size have an equal probability of being chosen; that is, simple random samples are used.

The samples are independent.

The conditions for a binomial distribution are met for both samples.

The sample sizes are large enough to ensure that n1pˆ1≥5

, n1(1−pˆ1)≥5, n2pˆ2≥5, and n2(1−pˆ2)≥5

.

When these conditions are met, we can apply the Central Limit Theorem to the sampling distribution of the differences between the sample proportions for two independent samples. This means that we will use the standard normal distribution to calculate the margin of error of a confidence interval for the difference between two population proportions. You can assume that the necessary criteria are met for all examples and exercises in this lesson.
Memory Booster

Population Proportion

p=xN=# of successespopulation size

Sample Proportion

pˆ=xn=# of successessample size

Properties of a Binomial Distribution

    The experiment consists of a fixed number, n, of identical trials.

    Each trial is independent of the others.

    For each trial, there are only two possible outcomes. For counting purposes, one outcome is labeled a success, and the other a failure.

    For every trial, the probability of getting a success is called p. The probability of getting a failure is then 1−p

.

The binomial random variable, X, counts the number of successes in n trials.

Confidence Interval Estimation

We have looked at creating a confidence interval for the difference between two population means using independent samples, meaning that the data from the two samples have no influence on each other. However, sometimes situations arise when the data sets are dependent. In this section, we will discuss how to construct a confidence interval for the difference between two population means using dependent samples where the observations in one sample uniquely correspond with observations in the second sample. Two dependent data sets, in which the observations from one data set are matched directly to the observations from the other data set, are called paired data.

So how do you decide when to design an experiment that will give you paired data? In general, you should select to use paired data when you want to compare two subgroups of a population that are logically connected. Each member of the first subgroup is systematically paired with a single member of the second subgroup either by matching characteristics or by using a preexisting connection, for example, twins. Here are some specific situations in which paired data would be used.

Pretest/posttest studies on the same subjects: For instance, suppose researchers wanted to study whether a person's sleeping habits changed when taking a new drug. Data would be taken from a number of participants both before the drug was administered and after. The data from each participant would then be paired together.

Pairing subjects with similar characteristics: The same research on sleep could occur by recruiting subjects as pairs by matching variables such as age, ethnicity, work environment, and so forth, and then giving one group a treatment (that is, the new drug) and the other a placebo.

Pairing subjects who have a specific connection that is of interest: For instance, parent/child pairings or sibling/twin pairings could reveal how certain genetic traits are related to patients' responses to the new drug.