Answer to Question #139048 in Statistics and Probability for Tuan

Question

What are the correct values of the mean and variance of the distribution of the heights of these students? 7. a. A box of biscuits is considered to be underweight if it weighs less than 228 grams. It is known that the weight of these boxes of biscuits is normally distributed with mean 231 grams and standard deviation 1.5 grams. What is the probability that the box is underweight?

Answer 1

Let $X=$ the weight of box of biscuits : $X\sim N(\mu, \sigma^2).$

Then $Z=\dfrac{X-\mu}{\sigma}\sim N(0, 1)$

Given $\mu=231g, \sigma=1.5g$

$$P(X<228)=P(Z<\dfrac{228-231}{1.5})=P(Z<-2)\approx$$

$$\approx0.022750$$

The probability that the box is underweight is approximately $0.022750.$

Bob measured the heights of 63 students. After analysis, he conjectured that the height, H, of the students could be modelled by a normal distribution with mean 166.5 cm and standard deviation 5 cm.

Later Bob noticed that the tape he had used to measure the heights was faulty as it started at the 5 cm mark and not at the zero mark.

What are the correct values of the mean and variance of the distribution of the heights of these students?

$$\mu_{new}=166.5cm-5cm=161.5cm$$

$$\sigma_{new}=5cm$$