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a. Find \(t_{05}\) for a \(t\) -distributed random variable with 5 df. b. Refer to part (a). What is \(P\left(T^{2}>t_{.05}^{2}\right) ?\) c. Find \(F_{10}\) for an \(F\) -distributed random variable with 1 numerator degree of freedom and 5 denominator degrees of freedom. d. Compare the value of \(F_{10}\) found in part \((c)\) with the value of \(t_{.05}^{2}\) from parts \((a)\) and \((b)\). e. In Exercise 7.33 , you will show that if \(T\) has a \(t\) distribution with \(\nu\) df, then \(U=T^{2}\) has an \(F\) distribution with 1 numerator degree of freedom and \(\nu\) denominator degrees of freedom. How does this explain the relationship between the values of \(F_{.10}\) (1 num. df, 5 denom df) and \(t_{05}^{2}(5 \mathrm{df})\) that you observed in part \((d)?\)

Step by step solution

01

Find t_{0.05} for a t-distributed random variable with 5 df

To find the critical value \( t_{0.05} \) for a \( t \)-distribution with 5 degrees of freedom, we need to determine the value such that the probability in the right tail is 0.05. This is typically found using a statistical table or software for the \( t \)-distribution. For a t-distribution with 5 degrees of freedom, \( t_{0.05} \approx 2.015 \).

02

Calculate P(T^{2}>t_{0.05}^{2})

We need to find the probability that \( T^{2} > t_{0.05}^{2} \). Since \( T^{2} \) follows a chi-squared distribution with 1 degree of freedom when \( T \sim t_{u} \) and \( u \) is 5, \( P(T^{2} > t_{0.05}^{2}) \) corresponds to a probability in the right tail of the chi-squared distribution with 1 degree of freedom. Because \( t_{0.05} \) has already been determined based on the right tail probability of 0.05, \( P(T^{2} > (2.015)^2) = 0.05 \).

03

Find F_{0.10} for an F-distributed random variable

To find \( F_{0.10} \) for an \( F \)-distribution with 1 numerator and 5 denominator degrees of freedom, we use an \( F \)-distribution table or software. \( F_{0.10} \) denotes the critical value such that the probability in the right tail is 0.10. We find that \( F_{0.10} \approx 4.9646 \).

04

Compare F_{0.10} and t_{0.05}^{2}

From steps (b) and (c), we found \( t_{0.05}^{2} \approx (2.015)^2 = 4.060225 \) and \( F_{0.10} \approx 4.9646 \). We compare these values: \( F_{0.10} > t_{0.05}^2 \).

05

Explain the relationship between F_{0.10} and t_{0.05}^{2}

In Exercise 7.33, it is shown that when \( T \) is \( t \)-distributed with \( u \) degrees of freedom, \( U = T^2 \) is \( F \)-distributed with 1 numerator and \( u \) denominator degrees of freedom. Thus, \( F_{0.10} \) from an \( F(1, 5) \) distribution relates directly to \( t_{0.05}^{2} \) because both involve the transformation to a chi-squared scale, yet the quantiles for \( F_{0.10} \) and \( t_{0.05}^2 \) differ due to different tail probabilities.

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Key Concepts

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

F-distribution

The F-distribution is a continuous probability distribution that arises frequently in statistical analysis, particularly in the analysis of variance (ANOVA), regression analysis, and hypothesis testing. It is crucial in situations where we want to compare two variances and is used to determine if they significantly differ from each other.

• The F-distribution is characterized by two types of degrees of freedom: the numerator degrees of freedom and the denominator degrees of freedom. These are usually represented as \( F(k_1, k_2) \), where \( k_1 \) and \( k_2 \) determine the shape of the distribution.
• It's positively skewed, with all its values being greater than zero, as it is constructed from the ratio of two chi-squared distributions divided by their respective degrees of freedom.
• The critical values obtained from the F-distribution (e.g., \( F_{0.10} \)) help in making decisions about statistical tests by comparing observed test statistics with these critical values.

In the case of an F-distribution with 1 numerator and 5 denominator degrees of freedom, finding \( F_{0.10} \) requires looking up tables or using statistical software that supplies the critical value where the right tail probability is 0.10.

degrees of freedom

Degrees of freedom (df) is a key concept in statistics that represents the number of independent values or quantities we can estimate in a statistical calculation.

• Degrees of freedom are used in various statistical tests, including the t-test, chi-squared test, and F-test, to determine the distribution that the test statistic follows.
• In the context of the t-distribution, degrees of freedom can be thought of as the sample size minus one (\( n - 1 \)) for a one-sample test, which influences the variability of the sampling distribution.
• For an F-distribution, as previously mentioned, there are two kinds of degrees of freedom: one for the numerator and another for the denominator, which come from the sources of variance being compared.

Understanding how degrees of freedom affect the shape of these statistical distributions is essential for accurately conducting and interpreting statistical analyses.

chi-squared distribution

The chi-squared distribution is a special case of the gamma distribution and is primarily used in statistical tests of independence and goodness-of-fit. It also plays a fundamental role in the derivation of the F-distribution.

• A chi-squared distribution with \( k \) degrees of freedom represents the sum of squares of \( k \) independent standard normal random variables.
• It’s a right-skewed distribution, becoming more symmetric as the degrees of freedom increase. The distribution is non-negative, similar to the F-distribution.
• In the context of our exercise, when a variable \( T \) follows a t-distribution, its square \( T^2 \) is distributed as a chi-squared distribution with 1 degree of freedom.

This aspect is important when assessing how the square of a t-distributed variable relates to the corresponding F-distribution.

right tail probability

Right tail probability refers to the probability that a statistical variable exceeds a certain value on the distribution. It’s an essential component of hypothesis testing.

• The right tail probability for a given distribution, like the t-distribution, chi-squared distribution, or F-distribution, helps in determining the critical values necessary for hypothesis testing.
• In hypothesis testing, we often calculate whether the test statistic falls into the right tail beyond the critical value to decide whether to reject the null hypothesis.
• In our exercise, finding \( t_{0.05} \) involved determining the critical value such that the probability in the right tail is 0.05 for the t-distribution with 5 degrees of freedom.

This concept is foundational in statistics, helping us decide the statistical significance based on calculated probabilities against set thresholds.