Problem 12 Find the \(t\) value(s) for each... [FREE SOLUTION]

Chapter 8: Problem 12

Find the \(t\) value(s) for each of the following cases. a. Upper tail area of .025 with 12 degrees of freedom b. Lower tail area of .05 with 50 degrees of freedom c. Upper tail area of .01 with 30 degrees of freedom d. Where \(90 \%\) of the area falls between these two \(t\) values with 25 degrees of freedom e. Where \(95 \%\) of the area falls between these two \(t\) values with 45 degrees of freedom

Short Answer

Expert verified

a. \(t = 2.1788\) b. \(t = -1.6759\) c. \(t = 2.7500\) d. \(t = \pm 1.7081\) e. \(t = \pm 2.0141\)

Step by step solution

Case a: Upper tail area of .025 with 12 degrees of freedom

1. Refer to the t-distribution table, and look for the row titled "12" under the column "DF" (degrees of freedom). 2. Now, locate the column that has ".025" in the row "One Tail". This is the t value that corresponds to the upper tail area of .025 with 12 degrees of freedom. 3. Write down the t value.

Case b: Lower tail area of .05 with 50 degrees of freedom

1. Refer to the t-distribution table, and look for the row titled "50" under the column "DF" (degrees of freedom). 2. Now, locate the column that has ".05" in the row "One Tail". This is the t value that corresponds to the lower tail area of .05 with 50 degrees of freedom. 3. Write down the t value.

Case c: Upper tail area of .01 with 30 degrees of freedom

1. Refer to the t-distribution table, and look for the row titled "30" under the column "DF" (degrees of freedom). 2. Now, locate the column that has ".01" in the row "One Tail". This is the t value that corresponds to the upper tail area of .01 with 30 degrees of freedom. 3. Write down the t value.

Case d: Where 90% of the area falls between these two t values with 25 degrees of freedom

1. Refer to the t-distribution table, and look for the row titled "25" under the column "DF" (degrees of freedom). 2. Since 90% of the area falls between these two t values, there is a 5% chance in each tail. So, locate the column that has ".05" in the row "One Tail". This is the t value that corresponds to this case. 3. Write down the t value.

Case e: Where 95% of the area falls between these two t values with 45 degrees of freedom

1. Refer to the t-distribution table, and look for the row titled "45" under the column "DF" (degrees of freedom). 2. Since 95% of the area falls between these two t values, there is a 2.5% chance in each tail. So, locate the column that has ".025" in the row "One Tail". This is the t value that corresponds to this case. 3. Write down the t value.

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Key Concepts

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

Degrees of Freedom

Degrees of freedom refers to the number of independent values in a dataset that are free to vary while calculating a statistic, like the mean or variance. In the context of the t-distribution, degrees of freedom (\( df \) ) generally equal the sample size minus one for a single sample.
This concept translates the size of your sample into a factor that influences the shape of the t-distribution curve. The formula often used is \( df = n - 1 \) , where \( n \) is the sample size.

Smaller degrees of freedom result in a t-distribution that is more spread out, resembling a normal distribution less closely.
As the degrees of freedom increase, the t-distribution becomes increasingly similar to a standard normal distribution.

Understanding degrees of freedom is essential when using t-tables, as each row in the table represents a different \( df \) and affects critical t-values.

Upper Tail Area

The upper tail area in a t-distribution represents the probability of observing a value greater than a given t-value. When you're looking at a t-table, the columns usually indicate probabilities for the upper tail 鈥� often labeled as "One Tail."
For instance, if you're asked for a t-value corresponding to an upper tail area of 0.025, you're seeking the t-value where there's a 2.5% chance of a value being larger, given the degrees of freedom of your data.

This is useful for hypothesis testing, especially for determining critical regions.
Finding this value typically involves identifying the correct degrees of freedom row and then locating the proper column showing your desired probability, like 0.025.

Understanding the upper tail area helps determine thresholds in statistical tests, where extreme values signal rare or significant outcomes.

Lower Tail Area

The lower tail area in a t-distribution signifies the probability of observing a value less than a specified t-value. Essentially the complement of the upper tail, it reflects the probability mass at the lower end of the distribution.
In practice, a lower tail area of 0.05 means there's a 5% chance of a t-value falling below that threshold given the degrees of freedom.

To find such a t-value, locate the row with the desired degrees of freedom in a t-table and the column showing your probability or "One Tail" cell.
It's often used in statistics to establish benchmarks for significance, supporting decisions like rejecting or accepting a null hypothesis.

Understanding lower tail areas allows statisticians to assess how likely data could fall under lower extremes, highlighting low-probability, significant results.

Confidence Interval

A confidence interval is a range of values used to estimate the true value of a population parameter, like the mean. When working with t-distributions, the confidence interval gives a sense of how certain we can be about a sample mean approximating the population mean.
A common question is "Where does 90% of the area of a t-distribution lie?" This involves locating t-values that leave a combined 10% in the tails 鈥� 5% in each tail if symmetrical鈥攗sing a specific degrees of freedom.

For a confidence level of 90%, look up the row for your degrees of freedom and find the column showing 0.05 in "One Tail" alignment.
A 95% confidence interval leaves 2.5% in each tail, guiding where to find corresponding t-values.

Knowing how to extract these values enables precise calculations and inferences, ensuring conclusions drawn from data are appropriately cautious.

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