To determine the value of [tex]\( z \)[/tex] to be used for calculating a confidence interval with a 90% confidence level, we need to understand the distribution of the data and the concept of the confidence level in statistics.
A confidence interval is a range of values, derived from the sample data, that is likely to contain the value of an unknown population parameter. The confidence level (in this case, 90%) represents how sure we are that the parameter lies within this interval.
For a 90% confidence interval, this means that there is 90% probability that the true population parameter will lie within this interval. Consequently, 10% of the distribution is in the tails, split equally with 5% in each tail.
[tex]\( z \)[/tex]-values represent points on the standard normal distribution curve (a bell curve) such that 90% of the area under the curve is within the interval centered at the mean. The remaining 10% is split into the tails.
To find the specific [tex]\( z \)[/tex]-value for a 90% confidence interval, we consult a standard normal distribution table or other statistical resources. According to the table provided in your question:
- [tex]\( z_{0.10} = 1.282 \)[/tex]
- [tex]\( z_{0.05} = 1.645 \)[/tex]
- [tex]\( z_{0.025} = 1.960 \)[/tex]
- [tex]\( z_{0.01} = 2.326 \)[/tex]
- [tex]\( z_{0.005} = 2.576 \)[/tex]
For a 90% confidence interval, we look at the [tex]\( z \)[/tex]-value corresponding to the 5% tail area under the standard normal curve, because 5% in each of the two tails adds up to 10%.
From the table above, the [tex]\( z \)[/tex]-value corresponding to [tex]\( 5% \)[/tex] in each tail (or 90% central probability) is:
[tex]\[
z_{0.05} = 1.645
\][/tex]
Therefore, the appropriate [tex]\( z \)[/tex]-value to use for a 90% confidence level is [tex]\( 1.645 \)[/tex]. So, your answer is:
[tex]\[
1.645
\][/tex]
