### Reducing Personal Call Expenses with a New Phone System**Introduction**A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $400 per month and a standard deviation of $50 per month.**Explanation**To understand the data, refer to such expenses as PCE’s (personal call expenses). Find the point in the distribution below which 2.5% of the PCE’s fell.**Details to Calculate**- **Mean (μ)**: $400 per month- **Standard Deviation (σ)**: $50 per month- **Distribution Type**: Normal Distribution**Calculation for the 2.5% Threshold**For a normal distribution, the point below which 2.5% of the values fall corresponds to approximately -1.96 standard deviations from the mean (using Z-scores).**Formula:**\[ X = \mu + Z \times \sigma \]Where:- $ \mu $ is the mean,- $ Z $ is the Z-score,- $ \sigma $ is the standard deviation,- $ X $ is the value to find.**Applying Values:**\[ X = 400 + (-1.96) \times 50 \]\[ X = 400 - 98 \]\[ X = 302 \]Therefore, below $302 is the point where 2.5% of personal call expenses fall under the previous phone system. This insight is crucial in understanding the behavior of personal call expenses and to gauge the effectiveness post-implementation of the new system.

From the given information we want to find the point in the distribution below which 2.5% of the…

A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $400 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the point in the distribution below which 2.5% of the PCE's fell.

A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $400 per month and a standard deviation of $50 per month. Refer to such expenses as PCE's (personal call expenses). Find the point in the distribution below which 2.5% of the PCE's fell.

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Concept explainers

Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!

Question

### Reducing Personal Call Expenses with a New Phone System

**Introduction**
A new phone system was installed last year to help reduce the expense of personal calls that were being made by employees. Before the new system was installed, the amount being spent on personal calls follows a normal distribution with an average of $400 per month and a standard deviation of $50 per month.

**Explanation**
To understand the data, refer to such expenses as PCE’s (personal call expenses). Find the point in the distribution below which 2.5% of the PCE’s fell.

**Details to Calculate**
- **Mean (μ)**: $400 per month
- **Standard Deviation (σ)**: $50 per month
- **Distribution Type**: Normal Distribution

**Calculation for the 2.5% Threshold**
For a normal distribution, the point below which 2.5% of the values fall corresponds to approximately -1.96 standard deviations from the mean (using Z-scores).

**Formula:**

\[ X = \mu + Z \times \sigma \]

Where:
- $ \mu $ is the mean,
- $ Z $ is the Z-score,
- $ \sigma $ is the standard deviation,
- $ X $ is the value to find.

**Applying Values:**

\[ X = 400 + (-1.96) \times 50 \]
\[ X = 400 - 98 \]
\[ X = 302 \]

Therefore, below $302 is the point where 2.5% of personal call expenses fall under the previous phone system.

This insight is crucial in understanding the behavior of personal call expenses and to gauge the effectiveness post-implementation of the new system.

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