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Question

A manufacturer knows that their items have a normally distributed lifespan, with a mean of 8.1 years, and standard deviation of 2.2 years.

If you randomly purchase 13 items, what is the probability that their mean life will be longer than 9 years?

A particular fruit's weights are normally distributed, with a mean of 407 grams and a standard deviation of 27 grams.

If you pick 14 fruit at random, what is the probability that their mean weight will be between 401 grams and 403 grams

Answer 1

1) We need to find the sample SD, s, from the population SD, so s=σ/√n=2.2/√13=0.61 approx. The deviation from the mean of 8.1 years is 9-8.1=0.9 years and we need to convert this into the number of sample SDs.

If we use the normal distribution test for 0.9/0.61=1.475, this corresponds to about 93%, which means that there is a 100-93=7% probability that the mean life will be longer than 9 years.

2) s=27/√14=7.216g approx. Let Z₁=(401-407)/7.216=-0.8315, Z₂=(403-407)/7.216=-0.5543.

These correspond to probabilities 20.28% and 28.97%, so the probability of the mean weight being between 401g and 403g is about 8.69%.

(Because of symmetry in the normal distribution, this is the same probability as between 411g and 413g (the "other side" of the mean of 407g).)