91Ó°ÊÓ

State which inference procedure from Chapter \(8,9,\) or 10 you would use. Be specific. For example, you might say, "Two-sample z test for the difference between two proportions." You do not need to carry out any procedures. Which inference method? (a) Drowning in bathtubs is a major cause of death in children less than 5 years old. A random sample of parents was asked many questions related to bathtub safety. Overall, \(85 \%\) of the sample said they used baby bathtubs for infants. Estimate the percent of all parents of young children who use baby bathtubs. (b) How seriously do people view speeding in comparison with other annoying behaviors? A large random sample of adults was asked to rate a number of behaviors on a scale of 1 (no problem at all) to 5 (very severe problem ). Do speeding drivers get a higher average rating than noisy neighbors? (c) You have data from interviews with a random sample of students who failed to graduate from a particular college in 7 years and also from a random sample of students who entered at the same time and did graduate. You will use these data to compare the percents of students from rural backgrounds among dropouts and graduates. (d) Do experienced computer game players earn higher scores when they play with someone present to cheer them on or when they play alone? Fifty teenagers with experience playing a particular computer game have volunteered for a study. We randomly assign 25 of them to play the game alone and the other 25 to play the game with a supporter present. Each player's score is recorded.

Step by step solution

01

Identify Inference Method for Problem (a)

For problem (a), we are estimating a single proportion. The appropriate inference procedure is a **one-sample z-interval for a proportion**. This method will estimate the percent of all parents who use baby bathtubs based on the sample proportion of 85%.

02

Identify Inference Method for Problem (b)

In problem (b), we want to compare the average ratings of speeding drivers and noisy neighbors based on a sample. This requires comparing two means. The appropriate inference method is a **paired t-test** (or **matched pairs t-test**) because we are comparing the mean difference in ratings for the same subjects.

03

Identify Inference Method for Problem (c)

In problem (c), the aim is to compare the proportions of students from rural backgrounds between two independent groups (dropouts and graduates). The appropriate inference method is a **two-sample z-test for the difference between two proportions**.

04

Identify Inference Method for Problem (d)

For problem (d), we are comparing two means from two independent groups (those who play with a supporter and those who play alone). The proper inference method is a **two-sample t-test for the difference between means**.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

One-Sample Z-Interval

The One-Sample Z-Interval is a statistical method used to estimate a population proportion based on a sample proportion. In simpler terms, it helps us understand what percentage of a population might have a certain characteristic, using information from a smaller group.
This method is particularly useful when dealing with a large sample size and when the sample is representative of the population. For example, when trying to understand how many parents use baby bathtubs for their infants, a One-Sample Z-Interval can provide an estimate for the wider population, based on the percentage found in a random sample.

• Calculation details: To calculate the Z-Interval, you start by determining the sample proportion. Then, using a Z-score (linked to the desired confidence level) and the standard error of the proportion, you can find the interval within which the true proportion likely falls.


• Key assumptions: This method assumes that the sample data are approximately normally distributed and that you have more than 30 observations in your sample.

This procedure provides a range, with an upper and lower bound, offering a clearer picture of where the true population proportion might lie.

Paired T-Test

A Paired T-Test is designed to compare two related means. It's particularly effective when you have paired data — like before-and-after measurements on the same subjects.
In the context of comparing ratings for speeding drivers and noisy neighbors, a Paired T-Test can be used. This is because respondents rate both behaviors, providing a pair of data points for each person.

• Steps involved: The first step in a Paired T-Test is to calculate the difference for each pair of observations. These differences are then analyzed to determine whether their average is significantly different from zero, using a t-distribution.


• Assumptions to consider: This test assumes that the differences among pairs are normally distributed and the pairs themselves are dependent, or related, in some way.

By understanding how the two means relate within the paired data, this test offers insights into whether significant differences exist.

Two-Sample Z-Test

The Two-Sample Z-Test compares the proportions of two independent groups. When researchers are interested in whether two different populations have different characteristic percentages, this test is an ideal choice.
For example, examining the difference in the proportion of students from rural areas among those who graduate and those who do not can be effectively addressed with a Two-Sample Z-Test.

• Key process: The method begins by identifying the two sample proportions. You then calculate the pooled proportion, which combines both sample proportions into a single estimate.


• Primary assumption: This test assumes that the samples are independent and random, and that each sample is sufficiently large for the Z-approximation (typically n > 30 for both samples).

This method gives a statistical basis to assess if there's a significant difference between the two groups' proportions, letting you know if any observed variation is likely due to random sampling error or an actual difference.

Two-Sample T-Test

The Two-Sample T-Test is utilized to compare the means of two independent groups. It's a go-to method when you want to see if the average from one group significantly differs from another.
In studies involving video game performance, for example, players might perform with or without a supporter present. Comparing the average scores across these groups can be effectively done with a Two-Sample T-Test.

• Execution steps: Begin by calculating the test statistic, using the difference of the sample means, pooled standard deviations, and the sample sizes. You then compare this statistic to a critical value from the t-distribution table.


• Crucial assumptions: This test assumes both samples are independent, drawn from populations that are approximately normally distributed, and the two samples have equal variances (homogeneity of variance).

Through this comparison, the Two-Sample T-Test identifies whether the difference in group means is significant, helping to determine if a factor (like having a supporter) impacts the result.