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Chapter 6: Problem 80

Find the area in a t-distribution above 2.3 if the sample has size \(n=6\).

Short Answer

Expert verified
To give an accurate short answer, the t-distribution table would need to be referenced, which is not available in this scenario. However, with the correct table, the steps provided will guide the student to the correct answer: 1 minus the table value associated with t=2.3 under 5 degrees of freedom.

Step by step solution

01

Find the Degrees of Freedom

The formula for degrees of freedom for a sample size \(n\) is \(df = n - 1\). This means with a sample size of \(n = 6\), the degrees of freedom \(df\) are \(6 - 1 = 5\).
02

Locate the Value in the Distribution Table

Now that the degrees of freedom have been calculated, these data can be used to find the value in the t-distributions table. It is critical to understand that tables usually give the area in the tail from the t-score to infinity, so the value of the area the table provides must be subtracted from 1 to get the area above the mentioned t-score.
03

Calculate the Area

Referencing a standard t-distribution table, under 5 degrees of freedom and a t-score of 2.3 (or the nearest t-score available), the given value will be the area in the tail from 2.3 to infinity. To find the area above 2.3, subtract the table value from 1. The result is the area in a t-distribution above 2.3.

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Key Concepts

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

Degrees of Freedom
The concept of "degrees of freedom" (often abbreviated as "df") is essential when working with statistical data, particularly in t-distributions. It tells us how many values in a data set are free to vary when calculating a statistic.
For example, if you have a sample size of 6, the degrees of freedom are calculated using the formula: \[ df = n - 1 \]where \( n \) is the sample size.
This means with a sample size \( n = 6 \), you have \( df = 5 \). By reducing the sample size by one, we account for the constraint imposed by estimating parameters like the sample mean.
Understanding degrees of freedom is crucial because it affects the shape of your t-distribution, influencing how you interpret your statistical results.
T-Score
A t-score is a type of standard score that indicates how much a data point diverges from the mean, measured in terms of standard deviation.
In simple terms, a t-score helps us understand the position of a data point within a t-distribution. This is particularly useful when dealing with small sample sizes.
To calculate the t-score, you use the formula: \[ t = \frac{\bar{x} - \mu}{s/\sqrt{n}} \]Where:
  • \( \bar{x} \) is the sample mean.
  • \( \mu \) is the population mean.
  • \( s \) is the sample standard deviation.
  • \( n \) is the sample size.
A higher t-score means the data point is further away from the mean, which may indicate statistical significance. T-scores are essential for looking up probabilities in a t-distribution table later.
Statistical Tables
Statistical tables, such as the t-distribution table, are tools that help find probabilities or critical values related to t-scores.
These tables provide the area under the t-distribution curve, frequently representing the probability of observing a value more extreme than the one calculated.
When using a t-distribution table:
  • Locate your calculated degrees of freedom in the left-hand column.
  • Move across the row to find the column corresponding to your calculated t-score (or the closest available score).
The value found in the table usually represents the area in the tail from the t-score to infinity.
In some cases, like finding the area above a t-score, you may need to subtract the table value from 1, giving the probability or the area under the curve beyond the t-score in question.

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Most popular questions from this chapter

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