Question

1. given: t - value = 2.492

find the following:
a) sample size n
b) df
c) $\alpha$
d) percentile

Answer:

To solve this, we refer to the t - distribution table. We know that for a t - value, we can find the degrees of freedom (df), and then relate it to the sample size \(n\), significance level \(\alpha\), and the percentile.

Part (a): Sample size \(n\)

Step 1: Recall the relationship between \(n\) and \(df\)

The formula for the degrees of freedom (\(df\)) for a one - sample t - test is \(df=n - 1\). So, if we find \(df\), we can find \(n\) by using the formula \(n=df + 1\).

Step 2: Find \(df\) from the t - table

We look up the t - value \(t = 2.492\) in the t - distribution table. We find that when \(t=2.492\), the degrees of freedom \(df = 24\) (this is a common value from the t - table for a two - tailed or one - tailed test, and we assume a two - tailed test for a general case here, or we can also consider the context of a one - sample or two - sample test. For a two - tailed test with \(\alpha = 0.02\) or a one - tailed test with \(\alpha=0.01\), but we will first find \(df\) from the t - value).

Step 3: Calculate \(n\)

Since \(df=n - 1\), and \(df = 24\), then \(n=df + 1=24 + 1=25\)

Part (b): Degrees of freedom (\(df\))

Step 1: Look up the t - value in the t - table

We search for the t - value \(t = 2.492\) in the t - distribution table. We find that the t - value \(t = 2.492\) corresponds to \(df=24\) (for example, in a two - tailed t - table, for \(\alpha = 0.02\), the t - value for \(df = 24\) is approximately \(2.492\); in a one - tailed t - table, for \(\alpha=0.01\), the t - value for \(df = 24\) is also approximately \(2.492\)).

Part (c): Significance level \(\alpha\)

Step 1: Determine the type of test (one - tailed or two - tailed)

If we assume a two - tailed test:
We know that for \(df = 24\) and \(t = 2.492\), from the t - table, the two - tailed \(\alpha\) value corresponding to \(t = 2.492\) and \(df = 24\) is \(\alpha=0.02\) (because the area in both tails adds up to \(0.02\)).

If we assume a one - tailed test:
The one - tailed \(\alpha\) value corresponding to \(t = 2.492\) and \(df = 24\) is \(\alpha = 0.01\) (because the area in one tail is \(0.01\)).

Part (d): Percentile

Step 1: Recall the relationship between percentile and t - value

The percentile can be calculated based on the significance level. If we consider a two - tailed test with \(\alpha = 0.02\), the percentile is \(100\times(1-\alpha)=100\times(1 - 0.02)=98\)th percentile (for a two - tailed test, the middle area is \(1-\alpha\)). If we consider a one - tailed test with \(\alpha = 0.01\), the percentile is \(100\times(1-\alpha)=100\times(1 - 0.01) = 99\)th percentile. But if we assume a two - tailed test (a common case in many introductory statistics problems), and we know that the t - value is for a two - tailed test with \(\alpha=0.02\), the percentile is \(98\)th percentile (since the area below the upper critical value and above the lower critical value is \(1-\alpha = 0.98\)).

Final Answers:

a) The sample size \(n=\boxed{25}\)

b) The degrees of freedom \(df=\boxed{24}\)

c) If we assume a two - tailed test, \(\alpha=\boxed{0.02}\); if we assume a one - tailed test, \(\alpha=\boxed{0.01}\)

d) If we assume a two - tailed test with \(\alpha = 0.02\), the percentile is \(\boxed{98}\)th percentile; if we assume a one - tailed test with \(\alpha=0.01\), the percentile is \(\boxed{99}\)th percentile. (The most common case for a t - value of \(2.492\) with \(df = 24\) in an introductory statistics context is a two - tailed test with \(\alpha = 0.02\) and 98th percentile or a one - tailed test with \(\alpha=0.01\) and 99th percentile)