Statistics: Two Independent Samples

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Questions and Answers

What is the sample mean resting pulse rate for non-exercisers?

28 bpm

66 bpm

75 bpm (correct)

9.0 bpm

What is the sample standard deviation for exercisers?

8.6 (correct)

9.0

What is the degree of freedom used for the confidence interval calculation in this study?

50 (correct)

What multiplier t* is used for large samples (n1 + n2 = 60)?

2.048 Signup and view all the answers

What does the confidence interval of 4.435 bpm to 13.565 bpm represent?

The range of differences in mean resting pulse rates between non-exercisers and exercisers Signup and view all the answers

What conclusion can be drawn since 0 is not in the confidence interval?

A difference can be claimed in mean resting pulse rates Signup and view all the answers

Which statistical procedure can be used because of the large sample sizes involved?

t-procedure Signup and view all the answers

What is the total number of subjects in both samples combined?

60 Signup and view all the answers

Which group had a lower mean resting pulse rate?

Exercisers Signup and view all the answers

If using the conservative degrees of freedom method, what is the conservative df calculated?

28 Signup and view all the answers

What is required for two samples to be considered independent?

Both samples must be drawn from normally distributed populations. Signup and view all the answers

Which of the following statements about the populations from which the samples are drawn is true?

Both populations should ideally have similar shapes. Signup and view all the answers

How are paired samples different from independent samples?

Paired samples have individuals from both groups influencing each other. Signup and view all the answers

What must hold true about the shapes of data distributions in independent samples?

They should have a similar shape and no strong outliers. Signup and view all the answers

In which scenario would matching individuals in two samples violate independence?

When participants are matched based on similar characteristics. Signup and view all the answers

Which of the following is a key characteristic of two-independent samples?

Observations in one sample do not affect observations in the other. Signup and view all the answers

What is an essential requirement for the populations from which independent samples are drawn?

Must have similar shapes. Signup and view all the answers

Which of the following factors does NOT affect the independence of two samples?

Pilot testing or preliminary experiments. Signup and view all the answers

When conducting inference on means from two samples, what must be true about the variables measured?

They should be quantitative and measured in the same unit. Signup and view all the answers

What is the main assumption for the validity of t-procedures for two independent samples?

Each sample must be an independent random sample from the population of interest. Signup and view all the answers

When is it appropriate to use t-procedures for small sample sizes?

When both samples have a symmetric and single-peaked distribution without outliers. Signup and view all the answers

What is the conservative approach for calculating degrees of freedom (df) in independent samples?

df = the smaller of (n1 - 1) or (n2 - 1) Signup and view all the answers

In what scenario can t-procedures be used for large samples?

The distributions may be skewed as long as sample sizes are large. Signup and view all the answers

What characteristic of the data sets affects the decision to use t-procedures for medium sample sizes?

The presence of outliers or strong skewness. Signup and view all the answers

To ensure an accurate estimate using a full degrees of freedom approximation, what should be verified?

The validity of the conservative approach is not required. Signup and view all the answers

Which sample sizes allow using t-procedures with clearly skewed distributions?

Sample sizes must be roughly 40 or more. Signup and view all the answers

Which condition is least important for independent samples when using t-procedures?

Sample sizes should be equal. Signup and view all the answers

What can invalidate the use of t-procedures regardless of sample size?

Outliers in the dataset. Signup and view all the answers

What is a necessary condition for small sample sizes when data is skewed?

The distribution must be symmetric. Signup and view all the answers

Study Notes

Two-Independent-Sample Problems

Comparing the mean resting pulse rate of adult subjects who regularly exercise to those who do not.
We have two independent random samples from two distinct populations.
The two samples are assumed to be independent and can be of different sizes.
We assume both population distributions are Normal.
We can use t-procedures for two independent samples.

Conditions for Independent Samples

The assumption that each sample is an independent random sample from the population of interest is more important than the assumption that the two population distributions are Normal.
For small sample sizes (n1 + n2 < 15), each data set must be close to Normal.
For medium sample sizes (n1 + n2 ≥ 15), t procedures are okay except in the presence of outliers or strong skewness.
For large samples (n1 + n2 ≥ 40), t procedures can be used even for skewed distributions.

Confidence Interval for Independent Samples

The full df approximation gives more accurate results than the conservative approach.
When calculating by hand, use the conservative approach: df = the smaller of (n1 – 1) or (n2 – 1).
Leave the full df calculation for software to do.

Exercise and Pulse Rates Example

A study compared the mean resting pulse rates of adult subjects who regularly exercise to those who don't.
We can use the t-procedure since the sample sizes are large (n1 + n2 = 60).
The conservative df is 28.
The multiplier t* is 2.048.
The 95% confidence interval for the difference in population means is between 4.435 bpm and 13.565 bpm.
Since 0 is not in the confidence interval, we can claim there is a difference in the mean resting pulse rates between non-exercisers and exercisers.

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Description

This quiz covers the concepts related to two-independent-sample problems in statistics. It includes comparisons of mean resting pulse rates in different populations and explores conditions for using t-procedures. Test your understanding of sample independence, normality assumptions, and construction of confidence intervals.