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Chapter 9: Problem 34

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. \(10 \mathrm{df}\), between -1.81 and 1.81 b. \(10 \mathrm{df}\), between -2.23 and 2.23 c. 24 df, between -2.06 and 2.06 d. \(24 \mathrm{df}\), 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
The percentages are as follows: a. 95%, b. 98%, c. 95%, d. 99.5%, e. 0.5%, f. 0.25%, g. 2.5%

Step by step solution

01

Understanding the Variables

Identify the degrees of freedom and the region where the values should fall. For example, in part (a), the degrees of freedom is 10 and the region is between -1.81 and 1.81.
02

Using the t-Distribution Table

Use a standard table of the t-distribution to find the cumulative probability associated with the given region. Remember to consider the symmetrical nature of the t-distribution when looking up values. Looking up 1.81 (the positive value) in the table for 10 degrees of freedom, we find that the cumulative probability is 0.95.
03

Calculating the Percentage

The cumulative probability in t-table equals to the percentage of the time that values fall within the region. So, in part (a), values fall within the region from -1.81 to 1.81 about 95% of the time.
04

Applying the same steps to the other parts.

Always repeat the same steps for each part of the problem. Remember that for outside the interval or to the right or left of a value, you would need to subtract the table value from 1.

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Key Concepts

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

Degrees of Freedom
When working with t-distributions, one key factor to understand is the concept of degrees of freedom. They are crucial as they affect the shape of the distribution.
Degrees of freedom refer to the number of values in a calculation that can vary independently. Typically, we calculate degrees of freedom as the sample size minus one, noted as \(df = n - 1\). This is because you usually estimate one parameter (like the mean) from your data, restricting one freedom.
Understanding degrees of freedom allows us to use the correct t-distribution table, ensuring accurate statistical analysis. More degrees of freedom mean the t-distribution looks more like a normal distribution.
Cumulative Probability
Cumulative probability indicates the likelihood that a variable takes a value less than or equal to a particular value. This concept helps answer questions like, "What is the probability that a variable falls within a certain range?" In t-distributions, cumulative probability helps us find the probability associated with a t-value. For instance, if you want the probability that a value is between -1.81 and +1.81 for 10 degrees of freedom, you look up the cumulative probability of 1.81 in the table. Remember, for values outside a certain range, you need to adjust the cumulative probability. Typically, this involves subtracting the found cumulative probability from 1.
T-Distribution Table
The t-distribution table is a powerful tool in statistics. It's used to find the probability related to a specific t-value, taking into account the degrees of freedom. The table contains rows and columns:
  • Rows usually represent degrees of freedom.
  • Columns represent t-values or probabilities.
The values in the table usually indicate the cumulative probability for a given t-value. For perfect use, locate the intersection based on your degrees of freedom and the t-value in question. This will give the cumulative probability, which tells you the likelihood of observing a t-value as extreme. Since t-tables assume symmetry, they're useful for two-tailed tests.
Statistical Interval Analysis
Statistical interval analysis involves finding the probability that a random variable's value lies within, or outside, a certain interval. This kind of analysis is integral to making predictions and understanding data behavior:
  • You need the cumulative probability for the critical t-values of your interval.
  • Subtract probabilities when you need ranges outside the interval or for one tail only.
For example, if assessing the interval from -2.80 to 2.80 with 24 degrees of freedom, you find cumulative probabilities for both values. Subtracting this cumulative probability from 1 gives the probability for values outside this range. This method provides a clear picture of how data is distributed.

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Most popular questions from this chapter

In a study of 1710 schoolchildren in Australia (Herald Sun, October 27,1994 ), 1060 children indicated that they normally watch TV before school in the morning. (Interestingly, only \(35 \%\) of the parents said their children watched TV before school!) Construct a \(95 \%\) confidence interval for the true proportion of Australian children who say they watch TV before school. What assumption about the sample must be true for the method used to construct the interval to be valid?

USA Today (October 14, 2002) reported that \(36 \%\) of adult drivers admit that they often or sometimes talk on a cell phone when driving. This estimate was based on data from a sample of 1004 adult drivers, and a bound on the error of estimation of \(3.1 \%\) was reported. Assuming a \(95 \%\) confidence level, do you agree with the reported bound on the error? Explain.

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: \(\begin{array}{lllll}11.3 & 10.7 & 12.4 & 15\end{array}\) \(16.2 \quad 10.5\) \(\begin{array}{llll}11.4 & 11.0 & 10.7 & 12.0\end{array}\) 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

The article "Students Increasingly Turn to Credit Cards" (San Luis Obispo Tribune, July 21,2006\()\) reported that \(37 \%\) of college freshmen and \(48 \%\) of college seniors carry a credit card balance from month to month. Suppose that the reported percentages were based on random samples of 1000 college freshmen and 1000 college seniors. a. Construct a \(90 \%\) confidence interval for the proportion of college freshmen who carry a credit card balance from month to month. b. Construct a \(90 \%\) confidence interval for the proportion of college seniors who carry a credit card balance from month to month. c. Explain why the two \(90 \%\) confidence intervals from Parts (a) and (b) are not the same width.

Consumption of fast food is a topic of interest to researchers in the field of nutrition. The article "Effects of Fast-Food Consumption on Energy Intake and Diet. Quality Among Children" reported that 1720 of those in a random sample of 6212 U.S. children indicated that on a typical day, they ate fast food. Estimate \(p\), the proportion of children in the United States who eat fast food on a tvpical dav.
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