Let's solve the problem step by step using the properties of the normal distribution.

Part a) What proportion of ATM users take between 80 and 100 seconds?

Given:

• Mean $\mu = 88$ seconds
• Standard deviation $\sigma = 20$ seconds

We are looking for the proportion of users that take between 80 and 100 seconds. This is equivalent to finding the area under the normal curve between these two values.

To do this, we first calculate the z-scores for 80 and 100 seconds using the formula for the z-score:

$z = \frac{x - \mu}{\sigma}$

1. For 80 seconds: $z_1 = \frac{80 - 88}{20} = \frac{-8}{20} = -0.4$

2. For 100 seconds: $z_2 = \frac{100 - 88}{20} = \frac{12}{20} = 0.6$

Now, we find the cumulative probability for each of these z-scores from the standard normal distribution.

• The cumulative probability for $z_1 = -0.4$ is approximately 0.3446.
• The cumulative probability for $z_2 = 0.6$ is approximately 0.7257.

To find the proportion of users taking between 80 and 100 seconds, subtract the cumulative probability at $z_1$ from the cumulative probability at $z_2$:

$\text{Proportion} = 0.7257 - 0.3446 = 0.3811$

Thus, approximately 38.11% of ATM users take between 80 and 100 seconds.

Part b) Compute the 99th percentile of ATM use times

To find the 99th percentile, we are looking for the time $x$ such that 99% of users take less time than $x$.

From the standard normal distribution table, the z-score corresponding to the 99th percentile is approximately $z = 2.33$.

Using the z-score formula, we can solve for $x$:

$z = \frac{x - \mu}{\sigma}$

Substituting the known values:

$2.33 = \frac{x - 88}{20}$

Solving for $x$:

$x = 2.33 \times 20 + 88 = 46.6 + 88 = 134.6 \, \text{seconds}$

Thus, the 99th percentile of ATM use times is approximately 134.6 seconds.

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Would you like further details on these calculations or have any questions?

Related Questions:

1. How would the proportion change if the standard deviation were 25 seconds?
2. What is the probability that a user takes less than 70 seconds at the ATM?
3. How would you calculate the interquartile range for this ATM usage time?
4. If you sample 5 users, what is the probability that their average time is less than 90 seconds?
5. How can we use the normal distribution to approximate percentiles for other datasets?

Tip: The standard normal distribution table is essential when working with z-scores and probabilities in normal distributions. Keep it handy!