To find the z-scores that correspond to the given cumulative probabilities (from a standard normal distribution table or z-table), we search for the closest probability values in the body of the table. The z-table provides cumulative probabilities for z-values from the mean (0) to a given z-score.

Let's solve each:

(a) $z_{0.4279}$

• Look for the closest cumulative probability to 0.4279 in the z-table.
• The closest value to 0.4279 is 0.42785, corresponding to $z = 0.08$.

Thus, $z_{0.4279} = 0.08$.

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(b) $z_{0.3997}$

• Look for the closest cumulative probability to 0.3997.
• The closest value to 0.3997 is 0.39970, corresponding to $z = 0.25$.

Thus, $z_{0.3997} = 0.25$.

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(c) $z_{0.3790}$

• Look for the closest cumulative probability to 0.3790.
• The closest value to 0.3790 is 0.37898, corresponding to $z = 0.31$.

Thus, $z_{0.3790} = 0.31$.

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(d) $z_{0.4951}$

• Look for the closest cumulative probability to 0.4951.
• The closest value to 0.4951 is 0.49507, corresponding to $z = 0.01$.

Thus, $z_{0.4951} = 0.01$.

Summary of z-scores:

• $z_{0.4279} = 0.08$
• $z_{0.3997} = 0.25$
• $z_{0.3790} = 0.31$
• $z_{0.4951} = 0.01$

Would you like further details on any of the steps?

Relative questions:

1. How do you interpret a z-score in a normal distribution?
2. Can you explain the process of finding a z-score using a z-table?
3. What is the significance of cumulative probability in a normal distribution?
4. How does the z-score relate to standard deviations in a normal distribution?
5. Can you find the probability for a z-score of 0.5 using the z-table?

Tip: When using a z-table, round the probability to the nearest value to match the closest cumulative probability in the table.