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Examine the following hypothesis test with [tex]\( n=20, \bar{x}=70.6, s=7.8 \)[/tex], and [tex]\( \alpha=0.10 \)[/tex].

[tex]\[
\begin{array}{l}
H_0: \mu \leq 70 \\
H_A: \mu\ \textgreater \ 70
\end{array}
\][/tex]

If you tested the hypothesis above, what is the t-critical value? (Hint: Identify the t-critical value associated with the given alpha level [tex]\(\alpha\)[/tex] and hypothesis test. You do not need to calculate any test statistic for this problem.)

(Round to two decimal places as needed.)


Sagot :

To determine the t-critical value for the given hypothesis test, follow these steps:

1. Identify the parameters:
- Sample size ([tex]\( n \)[/tex]) = 20
- Significance level ([tex]\( \alpha \)[/tex]) = 0.10

2. Determine the degrees of freedom:
The degrees of freedom (df) for a t-distribution is calculated as:
[tex]\[ \text{df} = n - 1 \][/tex]
Substituting the given sample size:
[tex]\[ \text{df} = 20 - 1 = 19 \][/tex]

3. Identify the significance level for the one-tailed test:
Since we have a one-tailed test at the significance level [tex]\(\alpha = 0.10\)[/tex], we will find the t-critical value corresponding to [tex]\( \alpha \)[/tex] in the t-distribution table or using statistical tools.

4. Find the critical t-value:
For a one-tailed test with [tex]\(\alpha = 0.10\)[/tex] and df = 19, the t-critical value (which corresponds to the [tex]\(90^{th}\)[/tex] percentile) can be identified from t-distribution tables or statistical software.

Based on these inputs, the t-critical value is found to be:
[tex]\[ t_{critical} = 1.33 \][/tex]

Therefore, the t-critical value for this hypothesis test is [tex]\( \boxed{1.33} \)[/tex]. This value serves as the threshold to decide whether to reject the null hypothesis [tex]\( H_0 \)[/tex]. If the calculated test statistic exceeds this critical value, then there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis [tex]\( H_A \)[/tex].