Find the indicated critical value. z 0.09
The correct answer and explanation is :
The critical value for z₀.₀₉ corresponds to the z-score where the cumulative area to the right is 0.09 (or 9%). This means we need to find the z-score such that P(Z > z) = 0.09, which is equivalent to finding P(Z < z) = 1 – 0.09 = 0.91.
From the standard normal table (or using a statistical calculator), the z-score that corresponds to a cumulative probability of 0.91 is approximately:
[
z_{0.09} = 1.34
]
Explanation:
- Understanding the Critical Value
A critical value is the boundary that separates the rejection region from the non-rejection region in hypothesis testing. It helps determine whether a test statistic falls within a confidence interval or in a region where we would reject the null hypothesis. - Why Use a Z-Score?
A z-score represents the number of standard deviations a data point is from the mean in a standard normal distribution (mean = 0, standard deviation = 1). Z-scores are widely used in hypothesis testing, confidence intervals, and probability calculations. - How to Find the Critical Value
- We need the z-score that leaves 0.09 (9%) in the upper tail.
- The area to the left of this z-score is 1 – 0.09 = 0.91.
- Using a z-table or calculator, the z-score corresponding to 0.91 cumulative probability is 1.34.
- Interpretation
The critical value z₀.₀₉ = 1.34 means that 91% of the data falls below this value in a standard normal distribution. This is useful in hypothesis testing when determining significance levels.
Thus, the final answer is:
[
z_{0.09} = 1.34
]