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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 \(2.80\) 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 exact probabilities will depend on the specific t-distribution table used or the specific function/software used in calculating these values from the t distribution. Note the probabilities for values less than the mean are equivalent to 1 - the probability for values greater than the mean due to the symmetry of the t-distribution. Hence, it's easier to subtract from 1 when the value is greater than the mean.

Step by step solution

01

Understanding the t Distribution

For a variable that follows a t distribution, the percentage of time it takes given values depends on its degrees of freedom and specified intervals. Degrees of freedom in this case refer to the number of values in a statistical calculation that have the freedom to vary.
02

Finding the Probability for Given Intervals

The t distribution tables (or appropriate software) can be used to find probabilities associated with given t-values for a specified degree of freedom. For example, for 10 degrees of freedom and the interval (-1.81, 1.81), lookup in the t distribution table gives the probability. Note, this gives the cumulative probability from the left end of the distribution to our t-value, and hence checking for -1.81 and 1.81 and subtracting gives the required probability.
03

Repeating Process for Other Intervals

This process should be repeated for the other given intervals and degrees of freedom. It's important to know that for a t-distribution, probabilities to the left and right of the mean are symmetric.
04

Finding the Probability to the Right or Left of a Value

When asked for the percentage of times the value is to the right or left of a given value, look up the cumulative probability of the t-value in the t-distribution table. If it is to the left, then that is the answer. But if it is to right, subtract the cumulative probability 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 we talk about 'degrees of freedom' in statistics, we are referring to the number of independent pieces of information that go into the estimation of a parameter. In the context of the t distribution, the degrees of freedom correlate with the number of data points in a sample that are 'free' to vary. Imagine a classroom of 11 students taking a test, and you know the average score. If you have the scores of 10 students, you can calculate the 11th student's score to maintain that average. Here, the degrees of freedom would be 10.

In our exercise examples, the '10 df' signifies a scenario where 10 values can freely vary. As the degrees of freedom increase, the t distribution gets closer to the normal distribution. This is vital for understanding how sample size affects the spread and the reliability of estimations in hypothesis testing. The '24 df' therefore implies a larger sample size, which generally means the results are more reliably close to a normal distribution.
Probability
Probability is a way of quantifying the likelihood of an event occurring. In statistics, the probability of a particular event involves the relationship between favorable outcomes divided by all possible outcomes. When we relate this to a t distribution, we are often looking for the probability that a statistic is within a particular range. For instance, we may ask: 'What is the chance that a t value falls between -1.81 and 1.81?'

In the step-by-step solution, utilizing the t distribution table allows us to pinpoint the likelihood of our variable falling within certain intervals, given a specific degree of freedom. The symmetry of the t distribution also plays a crucial role. Due to its bell-shaped curve, we know that the probability of falling to the left or right of the mean is the same. This symmetry eases the calculation of probabilities for negative and positive values of the same magnitude.
t Distribution Table
The t distribution table is a staple in statistical analysis when dealing with small sample sizes and when the population variance is unknown. This table provides critical values for the t distribution based on the degrees of freedom and desired level of significance. To utilize the table effectively, a user locates the row corresponding to the degrees of freedom and then the column for the required probability or confidence level. The intersection of this row and column gives the t value.

For instance, with 10 degrees of freedom and looking for an interval from -1.81 to 1.81 as in part a of our exercise, the table reveals the cumulative probability for these t-values. Understanding the table and knowing how to use it aids students in conducting hypothesis testing, constructing confidence intervals, and more. Remember though, in today's digital age, software often replaces manual lookups, streamlining computations but understanding how to read this table is a foundational skill in statistics.

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Most popular questions from this chapter

Authors of the news release titled "Major Gaps Still Exist Between the Perception and the Reality of Americans' Internet Security Protections, Study Finds" (The National Cyber Security Alliance) estimated the proportion of Americans who claim to have a firewall installed on their computer to protect them from computer hackers to be \(.80\) based on a survey conducted by the Zogby market research firm. They also estimated the proportion of those who actually have a firewall installed to be .42, based on checkups performed by Norton's \(\mathrm{PC}\) Help software. The following quote is from the news release: For the study, NCSA commissioned a Zogby survey of more than 3000 Americans and Symantec conducted checkups of 400 Americans' personal computers performed by PC Help by Norton (www.norton.com/tuneup). The Zogby poll has a margin of error of \(+/-1.6 \%\) and the checkup has a margin of error of \(+/-5 \%\). Explain why the margins of error for the two estimated proportions are different.

The authors of the paper "Short-Term Health and Economic Benefits of Smoking Cessation: Low Birth Weight" (Pediatrics [1999]: \(1312-1320\) ) investigated the medical cost associated with babies born to mothers who smoke. The paper included estimates of mean medical cost for low-birth-weight babies for different ethnic groups. For a sample of 654 Hispanic lowbirth-weight babies, the mean medical cost was $$\$ 55,007.$$ and the standard error \((s / \sqrt{n})\) was $$\$ 3011 .$$ For a sample of 13 Native American low-birth- weight babies, the mean and standard error were $$\$ 73,418$$ and $$\$ 29,577,$$ respectively. Explain why the two standard errors are so different.

The article "Hospitals Dispute Medtronic Data on Wires" (The Wall Street journal. February 4,2010 ) describes several studies of the failure rate of defibrillators used in the treatment of heart problems. In one study conducted by the Mayo Clinic, it was reported that failures were experienced within the first 2 years by 18 of 89 patients under 50 years old and 13 of 362 patients age 50 and older who received a particular type of defibrillator. Assume it is reasonable to regard these two samples as representative of patients in the two age groups who receive this type of defibrillator. a. Construct and interpret a \(95 \%\) confidence interval for the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator. b. Construct and interpret a \(99 \%\) confidence interval for the proportion of patients age 50 and older who experience a failure within the first 2 years after receiving this type of defibrillator. c. Suppose that the researchers wanted to estimate the proportion of patients under 50 years old who experience a failure within the first 2 years after receiving this type of defibrillator to within \(.03\) with \(95 \%\) confidence. How large a sample should be used? Use the results of the study as a preliminary estimate of the population proportion.

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.

The authors of the paper "Deception and Design: The Impact of Communication Technology on Lying Behavior" (Proceedings of Computer Human Interaction [2004]) asked 30 students in an upper division communications course at a large university to keep a journal for 7 days, recording each social interaction and whether or not they told any lies during that interaction. A lie was defined as "any time you intentionally try to mislead someone." The paper reported that the mean number of lies per day for the 30 students was \(1.58\) and the standard deviation of number of lies per day was \(1.02 .\) a. What assumption must be made in order for the \(t\) confidence interval of this section to be an appropriate method for estimating \(\mu\), the mean number of lies per day for all students at this university? b. Would you recommend using the \(t\) confidence interval to construct an estimate of \(\mu\) as defined in Part (a)? Explain why or why not.
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