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Chapter 9: Problem 30

Given a variable that has a \(t\) distribution with the specified degrees of freedom, what percentage of the time will its value fall in the indicated region? a. 10 df, between \(-1.81\) and \(1.81\) b. 10 df, between \(-2.23\) and \(2.23\) c. 24 df, between \(-2.06\) and \(2.06\) d. 24 df, between \(-2.80\) and \(2.80\) e. 24 df, outside the interval from \(-2.80\) to \(2.80\) f. \(24 \mathrm{df}\), to the right of \(2.80\) g. \(10 \mathrm{df}\), to the left of \(-1.81\)

Short Answer

Expert verified
a. 90% b. 95% c. 96% d. 99% e. 1% f. 0.5% g. 5%

Step by step solution

01

Understand the problem

Each question is referring to a specific range in a t-distribution. For example, when asked '10 df, between -1.81 and 1.81', it's really asking: given a t-distribution with 10 degrees of freedom, within what range does the t-distribution value fall 95% of the time? To find this out, you look up the two-tailed t value that corresponds to a cumulative probability of 95% with 10 degrees of freedom, which is between -1.81 and 1.81.
02

Apply the concept

To answer each lettered question, one would look up the relevant t-distribution value given the degrees of freedom and the percentage of time required. For example, in (b), '10 df, between -2.23 and 2.23' indicates that 95% of the time, the values will fall within a range of -2.23 to 2.23 for a t-distribution with 10 degrees of freedom.
03

Answer the question

Simply state the percentage corresponding to the given t-distribution values and degrees of freedom.

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Key Concepts

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

Degrees of Freedom
Understanding degrees of freedom (df) is crucial in the context of the t-distribution, especially when calculating probabilities for statistical tests. Degrees of freedom typically refer to the number of independent values or quantities in a calculation.
  • In a t-distribution, degrees of freedom are often related to the sample size. For instance, if you have a sample size of n, the degrees of freedom is usually n-1.
  • Degrees of freedom are essential because they determine the shape of the t-distribution curve. The greater the degrees of freedom, the more the t-distribution resembles a standard normal distribution.
Hence in the given problem, we might have degrees of freedom such as 10 or 24, which informs us of how to look up values from statistical tables, which then aids us in calculating the probability of a t-distributed random variable falling within a particular interval or region.
Probability
Probability is a fundamental concept in statistics that measures the likelihood of an event occurring. When we're dealing with t-distributions, probability helps us understand how likely a t-value will fall within a specified range.
  • For example, probabilities are calculated based on areas under the curve of a t-distribution. These areas give us the likelihood of a value being below or above a certain threshold, or within a designated interval.
  • In our exercise, when we say 95% probability for a certain range, it means there is a 95% chance that a randomly chosen t-value will fall within that interval.
These probabilities are crucial because they allow statisticians to make inferences and decisions based on data. Understanding probability lets you interpret how often certain outcomes are expected to occur statistically.
Cumulative Probability
Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. In the context of t-distributions, this is particularly relevant.
  • When you refer to cumulative probability, you're often talking about the total area under the t-distribution curve to the left of a certain value.
  • This concept is essential when determining how much of the distribution falls below or above certain critical values.
In the exercise, cumulative probabilities help us assess what percentage of the data lies within specified bounds. For example, if the cumulative probability associated with a range is 0.95, then 95% of the distribution's data lies within that range, confining our t-values to those limits. Cumulative probabilities are leveraged frequently in hypothesis testing and statistical inference to understand data behaviour as a whole.
Two-Tailed Test
The phrase 'two-tailed test' in statistics refers to a type of hypothesis test. This test checks for the possibility of a relationship in both directions鈥攎eaning it can detect if a variable significantly exceeds or falls below the expected value.
  • In a two-tailed test, the critical region where an effect is considered statistically significant is split between two tails of the distribution.
  • This is important because it covers both extremes of the data spread鈥攅xtremely low or high values relative to what you'd expect if only random chance were at work.
In our context, when looking at intervals such as between \( -2.23 \) and \( 2.23 \), it reinforces the notion of equally examining both directions of deviation from a mean. Therefore, the probabilities you state for a two-tailed test are doubled, as you are considering deviations in both directions from the mean, crucial for research and testing that doesn't assume directionality before conducting the analysis.

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Most popular questions from this chapter

In a survey of 1000 randomly selected adults in the United States, participants were asked what their most favorite and what their least favorite subject was when they were in school (Associated Press, August 17,2005 ). In what might seem like a contradiction, math was chosen more often than any other subject in both categories! Math was chosen by 230 of the 1000 as the favorite subject, and it was also chosen by 370 of the 1000 as the least favorite subject. a. Construct a \(95 \%\) confidence interval for the proportion of U.S. adults for whom math was the favorite subject in school. b. Construct a \(95 \%\) confidence interval for the proportion of U.S. adults for whom math was the least favorite subject.

The use of the interval $$ p \pm(z \text { critical value }) \sqrt{\frac{p(1-p)}{n}} $$ requires a large sample. For each of the following combinations of \(n\) and \(p\), indicate whether the given interval would be appropriate. a. \(n=50\) and \(p=.30\) b. \(n=50\) and \(p=.05\) c. \(n=15\) and \(p=.45\) d. \(n=100\) and \(p=.01\) e. \(n=100\) and \(p=.70\) f. \(n=40\) and \(p=.25\) g. \(n=60\) and \(p=.25\) h. \(n=80\) and \(p=.10\)

The formula used to compute a confidence interval for the mean of a normal population when \(n\) is small is $$ \bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}} $$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(95 \%\) confidence, \(n=17\) b. \(90 \%\) confidence, \(n=12\) c. \(99 \%\) confidence, \(n=24\) d. \(90 \%\) confidence, \(n=25\) e. \(90 \%\) confidence, \(n=13\) f. \(95 \%\) confidence, \(n=10\)

A manufacturer of college textbooks is interested in estimating the strength of the bindings produced by a particular binding machine. Strength can be measured by recording the force required to pull the pages from the binding. If this force is measured in pounds, how many books should be tested to estimate with \(95 \%\) confidence to within \(0.1 \mathrm{lb}\), the average force required to break the binding? Assume that \(\sigma\) is known to be \(0.8 \mathrm{lb}\).

The article "Most Canadians Plan to Buy Treats, Many Will Buy Pumpkins, Decorations and/or Costumes" (Ipsos-Reid, October 24, 2005) summarized results from a survey of 1000 randomly selected Canadian residents. Each individual in the sample was asked how much he or she anticipated spending on Halloween during 2005 . The resulting sample mean and standard deviation were \(\$ 46.65\) and \(\$ 83.70\) respectively. a. Explain how it could be possible for the standard deviation of the anticipated Halloween expense to be larger than the mean anticipated expense.
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