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Cloud [144]
3 years ago
9

A firm’s marketing manager believes that total sales for next year will follow the normal distribution, with a mean of $3.2 mill

ion and a standard deviation of $250,000. Determine the sales level that has only a 3% chance of being exceeded next year.
Mathematics
1 answer:
klasskru [66]3 years ago
7 0

Answer:

The sales level that has only a 3% chance of being exceeded next year is $3.67 million.

Step-by-step explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean \mu and standard deviation \sigma, the zscore of a measure X is given by:

Z = \frac{X - \mu}{\sigma}

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

In millions of dollars,

\mu = 3.2, \sigma = 0.25

Determine the sales level that has only a 3% chance of being exceeded next year.

This is the 100 - 3 = 97th percentile, which is X when Z has a pvalue of 0.97. So X when Z = 1.88.

Z = \frac{X - \mu}{\sigma}

1.88 = \frac{X - 3.2}{0.25}

X - 3.2 = 0.25*1.88

X = 3.67

The sales level that has only a 3% chance of being exceeded next year is $3.67 million.

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