Assume that 99.7% of grade 11 students have weights between 45 kg and 60 kg and the data are normally distributed. a)find the mean b) compute the standard deviation c) construct the normal curve of the normal distribution

Let $X$ denote the weights of grade 11 students.

Given that $p(45\lt x\lt60)=0.997$, we use this information to solve for the parameters $\mu$ and $\sigma.$   

To solve this, we shall apply the empirical rule. It states that, 99.7% of the data observed following a normal distribution lies within 3 standard deviations of the mean.

3 standard deviations below the mean can be written as, $\mu-3\sigma$ and is equal to the lower limit. For our case, the lower limit is 45. Thus, $\mu-3\sigma=45.........(i)$

3 standard deviations above the mean can be written as, $\mu+3\sigma$ and is equal to the upper limit. For our case, the upper limit is 60. Thus, $\mu+3\sigma=60.........(ii)$.

$a)$

To solve for the mean, we use equations $i)$ and $ii)$.

Adding equations $i)$ and $ii)$ gives,

$2\mu=105\implies \mu=52.5$

$b)$

To solve for the standard deviation, we use equations $i)$ and $ii)$.

Subtracting equation $i)$ from equation $ii)$ gives,

$6\sigma=15\implies \sigma=2.5$

$c)$

The normal curve of the normal distribution is given below.