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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

Each person in a random sample of 20 students at a particular university was asked whether he or she is registered to vote. The responses \((\mathrm{R}=\) registered, \(\mathrm{N}=\) not registered) are given here: R R N R N N R R R N R R R R R N R R R N Use these data to estimate \(\pi\), the true proportion of all students at the university who are registered to vote.

A random sample of \(n=12\) four-year-old red pine trees was selected, and the diameter (in inches) of each tree's main stem was measured. The resulting observations are as follows: $$ \begin{array}{llllllll} 11.3 & 10.7 & 12.4 & 15.2 & 10.1 & 12.1 & 16.2 & 10.5 \\ 11.4 & 11.0 & 10.7 & 12.0 & & & & \end{array} $$ a. Compute a point estimate of \(\sigma\), the population standard deviation of main stem diameter. What statistic did you use to obtain your estimate? b. Making no assumptions about the shape of the population distribution of diameters, give a point estimate for the population median diameter. What statistic did you use to obtain the estimate? c. Suppose that the population distribution of diameter is symmetric but with heavier tails than the normal distribution. Give a point estimate of the population mean diameter based on a statistic that gives some protection against the presence of outliers in the sample. What statistic did you use? d. Suppose that the diameter distribution is normal. Then the 90 th percentile of the diameter distribution is \(\mu+\) \(1.28 \sigma\) (so \(90 \%\) of all trees have diameters less than this value). Compute a point estimate for this percentile. (Hint: First compute an estimate of \(\mu\) in this case; then use it along with your estimate of \(\sigma\) from Part (a).)

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 Chronicle of Higher Education (January 13 , 1993) reported that \(72.1 \%\) of those responding to a national survey of college freshmen were attending the college of their first choice. Suppose that \(n=500\) students responded to the survey (the actual sample size was much larger). a. Using the sample size \(n=500\), calculate a \(99 \%\) confidence interval for the proportion of college students who are attending their first choice of college. b. Compute and interpret a \(95 \%\) confidence interval for the proportion of students who are not attending their first choice of college. c. The actual sample size for this survey was much larger than 500 . Would a confidence interval based on the actual sample size have been narrower or wider than the one computed in Part (a)?

The article "Viewers Speak Out Against Reality TV" (Associated Press, September 12,2005\()\) included the following statement: "Few people believe there's much reality in reality TV: a total of 82 percent said the shows are either 'totally made up' or 'mostly distorted'." This statement was based on a survey of 1002 randomly selected adults. Compute and interpret a bound on the error of estimation for the reported percentage.
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