Final answer:
The z-value that corresponds to the bottom 30% of the standard normal distribution is approximately -0.524. This value is found by looking up the area to the left in a z-table or using statistical software, and it indicates that the z-value is to the left of the mean for the lower percentage of a distribution.
Step-by-step explanation:
To find the z-value that corresponds to the bottom 30% of the standard normal distribution, one would look up the z-score that has 30% of the area under the curve to its left. This is often found using a z-table or a statistical software. In reference to the given information, we know that to capture the central 90%, we must go out 1.645 standard deviations from the mean; however, since we are looking for the bottom 30%, we will need to find a z-score that is less than zero.
According to typical z-tables, the z-score corresponding to an area of 0.30 to the left is approximately -0.524. The negative sign indicates that this z-value is to the left of the mean, which is typical for the lower percentage of a distribution.
It is important to note that the z-scores for intervals of the standard normal distribution are as follows:
- About 68 percent of the values lie between z-scores of -1 and 1.
- About 95 percent of the values lie between z-scores of -2 and 2.
- About 99.7 percent of the values lie between z-scores of -3 and 3.
These rules are part of what's known as the empirical rule or the 68-95-99.7 rule, and although they provide a quick way to estimate probabilities for certain intervals, the exact z-score for a specific percentile like 30% must be looked up individually or calculated using statistical software. In this case, the z-score we are looking for is stated as approximately -0.524 to answer the student's question.