Question

If the sample size for each treatment is \(n_{t}\) and \(s^{2}=8.0\) is based on \(k\left(n_{t}-1\right)\) degrees of freedom, find \(\omega=q_{\alpha}(k, d f)\left(\frac{s}{\sqrt{n_{t}}}\right)\) using the information. $$ \alpha=.01, k=6, n_{t}=6 $$

Short answer

Based on the information provided and using the given formula, we calculated that the value of \(\omega\) is approximately 7.16.

Step-by-step solution

The degrees of freedom can be calculated using \(k\left(n_{t}-1\right)\) $$ df=k(n_{t}-1)=6(6-1)=6(5)=30 $$ The degrees of freedom is 30. #Step 2: Find the \(q_{\alpha}(k, df)\) value#

To find the \(q_{\alpha}(k, df)\) value, we can refer to a Studentized Range Distribution table or use statistical software. Given the \(\alpha=0.01\), \(k=6\), and \(df=30\), the estimated value of \(q_{\alpha}(6, 30)\) is approximately 6.2. #Step 3: Calculate the standard deviation #

We are given the variance \(s^2 = 8.0\), and we need to find the standard deviation \(s\). The standard deviation is the square root of the variance. $$ s = \sqrt{s^2} = \sqrt{8.0} \approx 2.83 $$ #Step 4: Calculate \(\omega\) using the given formula#

Now substitute all the values in the given formula to find \(\omega\): $$ \omega = q_{\alpha}(k, df)\left(\frac{s}{\sqrt{n_{t}}}\right) = 6.2\left(\frac{2.83}{\sqrt{6}}\right) = 6.2 \times \frac{2.83}{2.45} \approx 7.16 $$ The value of \(\omega\) is approximately 7.16.

Key concepts

Degrees of Freedom

Understanding degrees of freedom (df) is crucial when dealing with statistical analyses, particularly hypothesis testing and variance estimations. In simple terms, degrees of freedom refer to the number of independent values or quantities which can be assigned to a statistical distribution. Imagine you are at a dinner party with one slice of cake less than the number of guests. Each guest gets to pick a slice one by one. The last person has no choice but must take the last piece — their 'freedom' to choose has been constrained by the others' choices.

In the context of variance or standard deviation, degrees of freedom refer to the number of independent pieces of information available to estimate variability. When calculating a sample variance, one degree of freedom is lost because the mean (a calculated value from the data) is used in the variance calculation. Hence, for a sample of size \( n_t \), the degrees of freedom is \( k(n_t-1) \). In the exercise given, where \( k = 6 \) treatments and the sample size for each treatment \( n_t = 6 \), the degrees of freedom were calculated to be 30, encompassing the independent pieces of information for variability estimation.

Standard Deviation

Standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, while a high standard deviation indicates that the values are spread out over a wider range. It is the square root of the variance, providing a metric that is in the same unit as the data.

Step 3 from the solution shows the calculation of standard deviation from variance. The variance (\( s^2 = 8.0 \)) was provided, and the formula for standard deviation is \( s = \sqrt{s^2} \). Upon calculating, we got the standard deviation to be approximately 2.83. This means that on average, the data points differ from the mean by about 2.83 units. In experiments, this serves as a vital indicator of consistency in measurements or results.

Variance

Variance is another measure of dispersion, but unlike standard deviation, it is the squared average difference of each number from the mean. Consequently, it provides a measure of the data's spread that is not in the same unit as the data, which can sometimes be conceptually challenging. From a practical standpoint, variance is essential because it is used in many statistical calculations and it forms the basis from which standard deviation is derived.

Using the data from Step 2, it's shown how a variance of \( s^2 = 8.0 \) was given. Understanding that this represents the variance — specifically, the average of the squared differences from the mean — is core to many statistical methods, including the calculation of the Confidence Interval for the mean and in the Analysis of Variance (ANOVA). The concept of variance is tied closely with degrees of freedom, as the estimation of variance adjusts based on the number of independent pieces of information in a dataset.