Question

A Soft Drink Machine At A Local Fast Food Restaurant Can Be Calibrated So That It Dispenses An Average Of Μμ Ounces Per Cup. If The Ounces Of Soda Dispensed Are Normally Distributed With Standard Deviation 0.5 Ounces, Find A Value Of Μμ Such That 32-Ounce Cups Will Overflow Only 10% Of The Time. Round The Solution To One Decimal Place, If Necessary. Μ=

A) A soft drink machine at a local fast food restaurant can be calibrated so that it dispenses an average of μμ ounces per cup. If the ounces of soda dispensed are normally distributed with standard deviation 0.5 ounces, find a value of μμ such that 32-ounce cups will overflow only 10% of the time. Round the solution to one decimal place, if necessary.

μ= ounces

B) SAT college entrance test scores are known to be a normally distributed with a mean of 1057 and a standard deviation of 53. Determine the 50th percentile of all SAT scores. Round the solution to the nearest whole number, if necessary.

The 50th percentile is

C)An honor society on campus only accepts the top 13% of students (according to GPA). If the GPAs of all students on campus are known to be a normally distributed with a mean of 3.23 and a standard deviation of 0.33, determine the minimum GPA a student must achieve in order to be admitted to the honor society. Round the solution to two decimal places, if necessary.

A minimum GPA of ____ is required for admission to the honor society.

Answer 1

For the soft drink machine to overflow only 10% of the time, we calculate μ using the 90th percentile and a standard deviation of 0.5 ounces. The 50th percentile of SAT scores is the mean, 1057. For the honor society, we find the z-score for the 87th percentile (top 13%) and apply it to the mean of 3.23 and standard deviation of 0.33 to determine the minimum GPA.

Step-by-step explanation:

To find the value of μ (the mean) for the soft drink machine so that 32-ounce cups will overflow only 10% of the time, we need to work with a normal distribution where we know the standard deviation (σ) is 0.5 ounces. Since we are concerned with the upper end of the distribution (overflowing), we are effectively looking for the 90th percentile of the distribution (100% - 10% overflow).

We can use a z-table, calculator, or software to find the z-score that corresponds to the 90th percentile. This z-score can then be used in the formula μ = X - zσ, where X is the 32 ounces (the overflow point). We solve for μ to find the mean we should set the machine to.

For the SAT scores, the 50th percentile corresponds to the median, which is also the mean in a normal distribution. Therefore, the 50th percentile is 1057.

To find the minimum GPA required to be admitted to the honor society, we look for the z-score corresponding to the top 13% of a normal distribution. With a mean (μ) of 3.23 and a standard deviation (σ) of 0.33, we can use the z-score to solve for the minimum GPA using μ + zσ.