To determine conditions for the use of a [tex]$t$[/tex]-distribution and answer the specific question, let's go through the necessary steps. The [tex]$t$[/tex]-distribution is typically used when the sample size is small and the population standard deviation is unknown. In this case, sample statistics substitute for population parameters.
Here’s a detailed step-by-step explanation:
1. Sample Size: The sample size is given as [tex]\( n = 44 \)[/tex].
2. Sample Mean: The sample mean ([tex]\(\bar{x}\)[/tex]) is given as [tex]\( \bar{x} = 151.2 \)[/tex].
3. Degrees of Freedom: The degrees of freedom (df) for the sample is calculated as [tex]\( df = n - 1 = 44 - 1 = 43 \)[/tex].
4. Standard Deviation: The standard deviation ([tex]\(s\)[/tex]) is not explicitly provided, but you mentioned it could be something like [tex]\( s = 1.234 \)[/tex].
5. Condition for Using the [tex]$t$[/tex]-distribution:
- When the sample size [tex]\( n \)[/tex] is small (usually [tex]\( n < 30 \)[/tex]), or the population standard deviation is unknown and the central limit theorem doesn't guarantee normality.
- Since [tex]\( n = 44 \)[/tex], which is larger than 30, the central limit theorem suggests the distribution of the sample mean is approximately normal. But since the population standard deviation is unknown, we should still rely on the [tex]$t$[/tex]-distribution.
-Thus, the condition is to use the [tex]$t$[/tex]-distribution because the population standard deviation is unknown, and the sample size is reasonably large.
Combining these elements, we conclude:
- Given a sample size [tex]\( n = 44 \)[/tex].
- Sample mean [tex]\( \bar{x} = 151.2 \)[/tex].
- Degrees of freedom [tex]\( df = 43 \)[/tex].
The requirement for using a [tex]$t$[/tex]-distribution is met under these conditions, primarily since the population standard deviation ([tex]\(\sigma\)[/tex]) is unknown.
