Question

what is the approximate probability that in a random sample of 1000 individuals who have purchased fries at McDonald’s, at least 40 can taste the difference between the two oils? (2.5 pts.)

Answer 1

Complete Question

In response to concerns about nutritional contents of fast foods, McDonald's has announced that it will use a new cooking oil for its French fries that will decrease substantially trans fatty acid levels and increase the amount of more beneficial polyunsaturated fat. The company claims that only 3 out of 100 people can detect a difference in taste between the new and old oils. Assuming that this figure is correct (as a long-run proportion), what is the approximate probability that in a random sample of 1000 individuals who have purchased fries at McDonald's, at least 40 can taste the difference between the two oils? (2.5 pts.)

Answer:

The probability is
`P(\^ p \ge 0.04 ) = 0.03216`

Explanation:

From the question we are told that

The population proportion is
`p = (3)/(100) = 0.03`

The sample size is n = 1000

Generally the mean of this sampling distribution is
`\mu_(x) = 0.03`

Generally the standard deviation of this sapling distribution is mathematically evaluated as

`\sigma = \sqrt{(p (1 - p))/(n) }`

=>
`\sigma = \sqrt{(0.03 (1 - 0.03))/(1000) }`

=>
`\sigma = 0.0054`

Generally the sample proportion when the number of those that can taste the difference is 40 is mathematically represented as

`\^ p = (40)/(1000) = 0.04`

Generally the approximate probability that in a random sample of 1000 individuals who have purchased fries at McDonald's, at least 40(
`\^ p = 0.04`) can taste the difference between the two oils is mathematically represented as

`P(\^ p \ge 0.04 ) = 1 - P(\^ p < 0.04 )`

Here

`P(\^ p < 0.04 ) = P(( \^ p - \mu_(x))/(\sigma ) < (0.04 - 0.03)/(0.0054) )`

`(\^ p -\mu)/(\sigma )  =  Z (The  \ standardized \  value\  of  \ \^ p )`

=>
`P(\^ p < 0.04 ) = P(Z < 1.85 )`

From the z table the probability of (Z < 1.85 ) is

`P(Z < 1.85 ) = 0.96784`

So

`P(\^ p < 0.04 ) = 0.96784`

So

`P(\^ p \ge 0.04 ) = 1 - 0.96784`

=>
`P(\^ p \ge 0.04 ) = 0.03216`