Question

A particular fruit's weights are normally distributed, with a mean of 426 grams and a standard deviation of 37 grams. If you pick 9 fruit at random, what is the probability that their mean weight will be between 413 grams and 464 grams. Round to 4 decimal places. Answers obtained using exact z-scores or z-scores rounded to 2 decimal places are accepted.

Answer 1

Probability that their mean weight will be between 413 grams and 464 grams is 0.8521.

Step-by-step explanation:

We are given that a particular fruit's weights are normally distributed, with a mean of 426 grams and a standard deviation of 37 grams.

Also, you pick 9 fruit at random.

Let
`\bar X` = sample mean weight

The z-score probability distribution for sample mean is given by;

Z =
`(\bar X-\mu)/((\sigma)/(√(n) ) )` ~ N(0,1)

where,
`\mu` = population mean weight = 426 grams

`\sigma` = population standard deviation = 37 grams

n = sample of fruits = 9

The Z-score measures how many standard deviations the measure is away from the mean. After finding the Z-score, we look at the z-score table and find the p-value (area) 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.

So, probability that the mean weight of 9 fruits picked at random will be between 413 grams and 464 grams is given by = P(413 grams <
`\bar X` < 464 grams) = P(
`\bar X` < 464 grams) - P(
`\bar X`
`\leq` 413 grams)

P(
`\bar X` < 464 grams) = P(
`(\bar X-\mu)/((\sigma)/(√(n) ) )` <
`(464-426)/((37)/(√(9) ) )` ) = P(Z < 3.08) = 0.99896

P(
`\bar X`
`\leq` 413 grams) = P(
`(\bar X-\mu)/((\sigma)/(√(n) ) )`
`\leq`
`(413-426)/((37)/(√(9) ) )` ) = P(Z
`\leq` -1.05) = 1 - P(Z < 1.05)

= 1 - 0.85314 = 0.14686

{Now, in the z table the P(Z
`\leq` x) or P(Z < x) is given. So, the above probability is calculated by looking at the value of x = 3.08 and x = 1.05 in the z table which has an area of 0.99896 and 0.85314 respectively.}

Therefore, P(413 grams <
`\bar X` < 464 grams) = 0.99896 - 0.14686 = 0.8521

Hence, the probability that their mean weight will be between 413 grams and 464 grams is 0.8521.