常態分佈：練習題

In a specific manufacturing process, the volume of liquid dispensed into bottles follows a normal distribution. It is known that \(95\%\) of the bottles contain between \(490.2\text{ ml}\) and \(509.8\text{ ml}\). Assuming the distribution is centered at \(500\text{ ml}\), calculate the standard deviation \(\sigma\) of the process.

A machine fills jars with honey. The weight of honey in a jar is \(H \sim N(\mu, \sigma^2)\). It is found that \(10\%\) of jars contain less than \(495\text{ g}\) and \(5\%\) contain more than \(510\text{ g}\). Calculate the value of \(\sigma\) to two decimal places.

The weights of a large population of birds follow a normal distribution \(W \sim N(240, 40^2)\). A researcher defines "underweight" as being in the bottom \(2.5\%\) of the population. Using the standard properties of the normal distribution, what is the maximum weight a bird can have to be considered underweight?

A factory produces lightbulbs where the probability of a bulb being defective is \(p = 0.05\). A random sample of \(200\) bulbs is tested. Using a normal approximation with a continuity correction, calculate the probability that exactly \(12\) bulbs are defective.

A company produces metal rods with lengths that are normally distributed with mean \(\mu = 50\text{ cm}\) and standard deviation \(\sigma = 0.5\text{ cm}\). If a rod is selected at random, what is the probability that its length is between \(49.2\text{ cm}\) and \(50.8\text{ cm}\)?

The scores in a test are normally distributed with a mean of 60 and a standard deviation of 8. Using the properties of the normal distribution, estimate the percentage of students who scored between 44 and 76.

The weights of a large population of bags of flour follow a normal distribution with mean \(\mu\) and standard deviation \(\sigma\). Given that \(P(X < 145) = 0.0228\) and \(P(X > 165) = 0.1587\), use properties of the standard normal distribution to determine the values of \(\mu\) and \(\sigma\).

A manufacturer of breakfast cereal ensures that the weight of cereal in a '500g' box, \(X\) grams, is normally distributed with mean \(\mu = 505\) g and standard deviation \(\sigma = 4\) g.

(a) Find the probability that a randomly selected box contains less than the stated weight of 500g. (2 points)
(b) A box is considered 'underfilled' if it is in the bottom 2.5% of the distribution. Find the weight below which a box is considered underfilled. (2 points)

A company manufactures light bulbs. The lifetime of these bulbs, \(X\) hours, follows a normal distribution with a mean \(\mu\) and a standard deviation \(\sigma\).

It is found that 10% of the bulbs last more than 1500 hours, and 5% of the bulbs last less than 800 hours.

(a) Show that \(\mu\) and \(\sigma\) satisfy the equations \(1500 = \mu + 1.2816\sigma\) and \(800 = \mu - 1.6449\sigma\). (2 points)

(b) Calculate the value of \(\mu\) and the value of \(\sigma\) to one decimal place. (2 points)

(c) A random sample of 40 bulbs is tested. Using the values from part (b), find the probability that the sample mean lifetime of these 40 bulbs is less than 1150 hours. (2 points)