正態變量的標準化及標準正態分佈表的運用：练习题

A random variable \(X\) follows a normal distribution with mean \(50\) and variance \(16\). What is the \(z\)-score corresponding to the value \(X = 56\)?

Let \(Z\) be the standard normal variable. If \(P(Z < 1.5) = 0.9332\), find \(P(-1.5 < Z < 0)\).

A random variable \(X\) follows a normal distribution \(N(\mu, 4^2)\). Given that \(P(X < 12) = 0.1587\) and \(P(Z < 1) = 0.8413\) where \(Z\) is the standard normal variable, find the value of \(\mu\).

The weights of a certain species of fish are normally distributed with an unknown mean \(\mu\) kg and a standard deviation of \(0.8\) kg. If \(15\%\) of these fish weigh more than \(7.5\) kg, find the probability that a randomly selected fish weighs less than \(6.0\) kg.

The daily electricity consumption, X, (in kWh) of a household is normally distributed with a mean of $$25$$ kWh and a standard deviation of $$4$$ kWh.

(a) Find the probability that the daily electricity consumption is between $$20$$ kWh and $$30$$ kWh.

(b) The electricity company charges a penalty if the daily consumption exceeds $$k$$ kWh. If $$15\%$$ of the days incur a penalty, find the value of $$k$$ to one decimal place.

(c) In a month of $$30$$ days, how many days are expected to have a daily consumption less than $$18$$ kWh?

The lifespan of a certain brand of LED light bulbs, L, (in hours) is normally distributed with a mean of $$\mu$$ hours and a standard deviation of $$500$$ hours. It is known that $$12.51\\%$$ of the light bulbs have a lifespan less than $$9000$$ hours.

(a) Find the mean lifespan $$\mu$$ of this brand of LED light bulbs.

(b) A light bulb is considered 'long-lasting' if its lifespan is between $$10000$$ hours and $$11500$$ hours. Find the probability that a randomly selected light bulb is long-lasting.

(c) A shop buys a batch of $$15$$ of these LED light bulbs. Find the probability that exactly $$3$$ of them are long-lasting.