To find the area to the left of [tex]\( z = -2.34 \)[/tex], we need to use the standard normal distribution table or a statistical tool known to calculate cumulative probabilities for the standard normal distribution.
1. Understanding the Standard Normal Distribution (SND):
The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The z-score indicates how many standard deviations an element is from the mean.
2. Locating the z-score:
We have a z-score of [tex]\( -2.34 \)[/tex]. A negative z-score indicates that the value is below the mean of the distribution. Specifically, it is 2.34 standard deviations to the left of the mean.
3. Cumulative Probability:
The area to the left of a given z-score under the standard normal curve can be found in a cumulative probability table for the standard normal distribution, typically referred to as the z-table, or using an appropriate statistical function.
4. Using Z-Table or Statistical Tool:
When you consult such a table or tool for [tex]\( z = -2.34 \)[/tex], you read off the cumulative probability up to that point. This value represents the area to the left of the z-score [tex]\( -2.34 \)[/tex] on the standard normal distribution.
Hence, the exact area to the left of [tex]\( z = -2.34 \)[/tex] is 0.00964186994535833. This value represents the cumulative probability, and it means that approximately 0.96% of the data under a standard normal distribution lies to the left of [tex]\( z = -2.34 \)[/tex].
