A vending machine dispenses coffee into an eight-ounce cup. The amounts of coffee dispensed into the cup are normally distributed, with a standard deviation of 0.03 ounce. You can allow the cup to overflow 1% of the time. What amount should you set as the mean amount of coffee to be dispensed?

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Cheryl King · Accepted Answer

So,x=Ounces of cup=8&amp;nbsp;σ=Standard deviation=0.03&amp;nbsp;One-tenth of one percent (tenth) of the data values in the top 1% are larger than the cutoff threshold and thus&amp;nbsp;100%−1%=99%&amp;nbsp;are less than the cutoff threshold for the data.The z-score corresponding to an area of 99% or 0.99 or the nearest area will be found using the standard normal table in the appendix.The value 0.99 in the appendix&#039;s standard normal table is the closest to 0.9901, as noted.The value.9901, which has a z-score of 2.33, is given in the row beginning with 2.3 and the column beginning with.03.z=2.33&amp;nbsp;The z-score is defined as the value of x divided by the mean and the standard deviation, though, as we also know.z=x−μσ&amp;nbsp;μ=x−zσ&amp;nbsp;Solve to&amp;nbsp;μ&amp;nbsp;=8−(2.33)(0.03)&amp;nbsp;Substitute&amp;nbsp;=7.9301&amp;nbsp;As a result, the average volume of coffee to be delivered is 7.9301 ounces, meaning that the 8-ounce cup will typically overflow 1% of the time.

Juan Spiller · Accepted Answer

Step 1   The amounts of coffee dispensed into the cup follow normal distribution with standard deviation 0.03 ounces. Also, the vending machine dispenses coffee into an eight-ounce cup and the cup to overflow is 1% of the time.   The random variable x represents the amounts of coffee dispensed.   The cup to overflow is 1% of the time represents the area of 0.01 to the right of x.   Thus, P(X&amp;lt;x)=0.99   P(X−μσ&amp;lt;x−μσ)=0.99 (by standardizing)   P(Z&amp;lt;x−μσ)=0.99   Φ(x−μσ)=0.99   Find the value 0.99 from the body of the standard normal table. The value is approximately 2.33.   Step 2   Φ(2.33)=0.99.   Thus,   Φ(x−μσ)=Φ(2.33)   x−μσ=2.33   x=μ+2.33(σ)   Substitute μ=8 and σ=0.03.   x=8+2.33(0.03)   =8.0699   The value can be also obtained using the Excel formula, =NORM.INV(0.99,8,0.03) as 8.0698.   Thus, the amount that should be set as the mean amount of coffee to be dispensed is 8.07 ounces.

A vending machine dispenses coffee into an eight-ounce cup. The amount

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