Answer :
We begin by noting that the number of days between when a bill is sent and when the payment is made is approximately normally distributed with a mean of [tex]$25$[/tex] days and a standard deviation of [tex]$7$[/tex] days.
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Step 1: Approximately 95% of the data
For a normally distributed variable, approximately [tex]$95\%$[/tex] of the values fall within [tex]$2$[/tex] standard deviations of the mean. That is, the interval is given by
[tex]$$
\text{Mean} \pm 2 \times \text{Standard Deviation}.
$$[/tex]
Substitute the given values:
- Lower bound:
[tex]$$
25 - 2 \times 7 = 25 - 14 = 11 \text{ days}.
$$[/tex]
- Upper bound:
[tex]$$
25 + 2 \times 7 = 25 + 14 = 39 \text{ days}.
$$[/tex]
Thus, approximately [tex]$95\%$[/tex] of the customer accounts have the payment made between [tex]$11$[/tex] and [tex]$39$[/tex] days.
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Step 2: Estimating the percentage for days between [tex]$18$[/tex] and [tex]$32$[/tex]
Next, we estimate the percentage of accounts for which the number of days falls between [tex]$18$[/tex] and [tex]$32$[/tex]. We first convert these values to [tex]$z$[/tex]-scores using the formula
[tex]$$
z = \frac{\text{Value} - \text{Mean}}{\text{Standard Deviation}}.
$$[/tex]
For [tex]$18$[/tex] days:
[tex]$$
z = \frac{18 - 25}{7} = \frac{-7}{7} = -1.
$$[/tex]
For [tex]$32$[/tex] days:
[tex]$$
z = \frac{32 - 25}{7} = \frac{7}{7} = 1.
$$[/tex]
The probability that a normally distributed variable falls between [tex]$z = -1$[/tex] and [tex]$z = 1$[/tex] is approximately [tex]$68.27\%$[/tex].
Thus, about [tex]$68\%$[/tex] of the customer accounts have the payment made between [tex]$18$[/tex] and [tex]$32$[/tex] days.
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Final Answers:
(a) Approximately [tex]$95\%$[/tex] of the customer accounts have payment made between [tex]$11$[/tex] and [tex]$39$[/tex] days.
(b) Approximately [tex]$68\%$[/tex] of the customer accounts have payment made between [tex]$18$[/tex] and [tex]$32$[/tex] days.
---
Step 1: Approximately 95% of the data
For a normally distributed variable, approximately [tex]$95\%$[/tex] of the values fall within [tex]$2$[/tex] standard deviations of the mean. That is, the interval is given by
[tex]$$
\text{Mean} \pm 2 \times \text{Standard Deviation}.
$$[/tex]
Substitute the given values:
- Lower bound:
[tex]$$
25 - 2 \times 7 = 25 - 14 = 11 \text{ days}.
$$[/tex]
- Upper bound:
[tex]$$
25 + 2 \times 7 = 25 + 14 = 39 \text{ days}.
$$[/tex]
Thus, approximately [tex]$95\%$[/tex] of the customer accounts have the payment made between [tex]$11$[/tex] and [tex]$39$[/tex] days.
---
Step 2: Estimating the percentage for days between [tex]$18$[/tex] and [tex]$32$[/tex]
Next, we estimate the percentage of accounts for which the number of days falls between [tex]$18$[/tex] and [tex]$32$[/tex]. We first convert these values to [tex]$z$[/tex]-scores using the formula
[tex]$$
z = \frac{\text{Value} - \text{Mean}}{\text{Standard Deviation}}.
$$[/tex]
For [tex]$18$[/tex] days:
[tex]$$
z = \frac{18 - 25}{7} = \frac{-7}{7} = -1.
$$[/tex]
For [tex]$32$[/tex] days:
[tex]$$
z = \frac{32 - 25}{7} = \frac{7}{7} = 1.
$$[/tex]
The probability that a normally distributed variable falls between [tex]$z = -1$[/tex] and [tex]$z = 1$[/tex] is approximately [tex]$68.27\%$[/tex].
Thus, about [tex]$68\%$[/tex] of the customer accounts have the payment made between [tex]$18$[/tex] and [tex]$32$[/tex] days.
---
Final Answers:
(a) Approximately [tex]$95\%$[/tex] of the customer accounts have payment made between [tex]$11$[/tex] and [tex]$39$[/tex] days.
(b) Approximately [tex]$68\%$[/tex] of the customer accounts have payment made between [tex]$18$[/tex] and [tex]$32$[/tex] days.
