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Chapter 6: Problem 217

Use a t-distribution. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the area in a t-distribution less than -1.4 if the samples have sizes \(n_{1}=30\) and \(n_{2}=40\).

Short Answer

Expert verified
The exact area depends on the specific t-distribution table or calculator used. The conceptually correct answer is 'The area less than -1.4 in a t-distribution with 68 degrees of freedom'.

Step by step solution

01

Calculate the Degrees of Freedom

This is a two-sample t-test, so the degrees of freedom is calculated with the formula: \[ df = n_{1} + n_{2} - 2 \]. Substituting the given sample sizes into the formula gives: \[ df = 30 + 40 - 2 = 68 \].
02

Use t-Distribution Table

To find the area less than -1.4 in a t-distribution with 68 degrees of freedom, one needs to use a t-distribution table or calculator. Because the t-distribution is symmetric, the area less than -1.4 is the same as the area more than 1.4.
03

Look up the Area

Using a t-distribution table or calculator, one can look up the value with degrees of freedom of 68 and t-score of 1.4. In many t-distribution tables, the exact degrees of freedom might not be listed. If so, one can approximate by using the closest value available.
04

Find the Result

The area value found in the t-distribution table or calculator is the solution to the problem.

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Key Concepts

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

Degrees of Freedom in T-Distribution
Degrees of freedom, often abbreviated to "df," refer to the number of values in a calculation that are free to vary. In the context of a t-distribution, they are crucial for determining the shape of the distribution. The degrees of freedom affect the tails of the t-distribution 鈥� the fewer the degrees of freedom, the thicker the tails, which in turn affects the confidence intervals and p-values obtained from the test.

When carrying out a two-sample t-test, the formula for calculating the degrees of freedom is:
  • \[ df = n_1 + n_2 - 2 \]
where \( n_1 \) and \( n_2 \) are the sizes of the two samples. As a result, when working with larger sample sizes, the t-distribution approaches the standard normal distribution.

In our specific exercise, with sample sizes of 30 and 40, respectively, we find:
  • \( df = 30 + 40 - 2 = 68 \)
This gives us the necessary information to locate the appropriate row when using a t-distribution table to find probabilities or critical values.
Understanding the T-Statistic
The t-statistic is the calculated value that represents the ratio of the difference between the sample means to the variability of the samples. It is a standardized value that allows us to measure how far apart the sample means are from each other, against what we would expect if the null hypothesis was true.

Mathematically, the t-statistic is calculated through:
  • \[ t = \frac{\bar{x}_1 - \bar{x}_2}{s_p \sqrt{\frac{1}{n_1} + \frac{1}{n_2}}} \]
where \( \bar{x}_1 \) and \( \bar{x}_2 \) are the sample means, \( s_p \) is the pooled standard deviation of the samples, and \( n_1 \) and \( n_2 \) are the sample sizes.

The absolute value of the t-statistic can then be compared against critical values from a t-distribution to determine significance, allowing us to make inferences about the population means. A larger absolute t-statistic indicates a greater deviation from the null hypothesis, suggesting that observed effects are less likely due to random chance.
Two-Sample T-Test
The two-sample t-test is a statistical technique used to compare means from two different groups to determine if they are significantly different from each other. The main purpose of this test is to assess whether the observed differences between group means are likely to reflect actual differences in the populations, or if they might just be due to random variation.

In conducting the two-sample t-test, the steps are straightforward:
  • First, calculate the mean and standard deviation for each sample.
  • Determine the confidence level and degrees of freedom.
  • Compute the t-statistic using the formula for pooled variance.
  • Compare the t-statistic to the critical value from the t-distribution table with the appropriate degrees of freedom.
When interpreting the results, if the calculated t-statistic surpasses the critical t-value, we reject the null hypothesis, indicating a significant difference between the sample means.

In our exercise, because we are looking at differences with a t-statistic of -1.4, which falls within the range of a two-tailed test, we need to ensure that we interpret whether the observed difference is statistically significant based on tools such as a t-distribution table or a calculator.

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Most popular questions from this chapter

A young statistics professor decided to give a quiz in class every week. He was not sure if the quiz should occur at the beginning of class when the students are fresh or at the end of class when they've gotten warmed up with some statistical thinking. Since he was teaching two sections of the same course that performed equally well on past quizzes, he decided to do an experiment. He randomly chose the first class to take the quiz during the second half of the class period (Late) and the other class took the same quiz at the beginning of their hour (Early). He put all of the grades into a data table and ran an analysis to give the results shown below. Use the information from the computer output to give the details of a test to see whether the mean grade depends on the timing of the quiz. (You should not do any computations. State the hypotheses based on the output, read the p-value off the output, and state the conclusion in context.) $$ \begin{aligned} &\text { Two-Sample T-Test and Cl }\\\ &\begin{array}{lrrrr} \text { Sample } & \mathrm{N} & \text { Mean } & \text { StDev } & \text { SE Mean } \\ \text { Late } & 32 & 22.56 & 5.13 & 0.91 \\ \text { Early } & 30 & 19.73 & 6.61 & 1.2 \end{array} \end{aligned} $$ Difference \(=\mathrm{mu}\) (Late) \(-\mathrm{mu}\) (Early) Estimate for difference: 2.83 \(95 \%\) Cl for difference: (-0.20,5.86) T-Test of difference \(=0\) (vs not \(=\) ): T-Value \(=1.87\) P-Value \(=0.066 \quad \mathrm{DF}=54\)

We see } in the AllCountries dataset that the percent of the population living in rural areas is 8.0 in Argentina and 34.4 in Bolivia. Suppose we take random samples of size 200 from each country, and compute the difference in sample proportions \(\hat{p}_{A}-\hat{p}_{B},\) where \(\hat{p}_{A}\) represents the sample proportion living in rural areas in Argentina and \(\hat{p}_{B}\) represents the proportion of the sample that lives in rural areas in Bolivia. (a) Find the mean and standard deviation of the distribution of differences in sample proportions, \(\hat{p}_{A}-\hat{p}_{B}\) (b) If the sample sizes are large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. (c) Using the graph drawn in part (b), are we likely to see a difference in sample proportions as large in magnitude as -0.4 ? As large as \(-0.3 ?\) Explain.

Use a t-distribution. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the area in a t-distribution above 2.1 if the samples have sizes \(n_{1}=12\) and \(n_{2}=12\).

In Exercises 6.1 to \(6.6,\) if random samples of the given size are drawn from a population with the given proportion: (a) Find the mean and standard error of the distribution of sample proportions. (b) If the sample size is large enough for the Central Limit Theorem to apply, draw a curve showing the shape of the sampling distribution. Include at least three values on the horizontal axis. Samples of size 60 from a population with proportion 0.90

In the dataset ICUAdmissions, the variable Infection indicates whether the ICU (Intensive Care Unit) patient had an infection (1) or not (0) and the variable Sex gives the gender of the patient \((0\) for males and 1 for females.) Use technology to test at a \(5 \%\) level whether there is a difference between males and females in the proportion of ICU patients with an infection.
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