Chapter 7.2, Problem 28E

Baby weights: According to a recent National Health Statistics Reports: the weight of male babies less than 2 months old in the United States is normally distributed with mean 11.5 pounds and standard deviation 2.7 pounds.

1. What proportion of babies weigh more than 13 pounds?
2. What proportion of babies weigh less than 15 pounds?
3. What proportion of babies weigh between 10 and 14 pounds?
4. Is it unusual for a baby to weigh more than 17 pounds?

To determine

To find: The proportion of baby weighs more than $13$ pounds.

Answer to Problem 28E

The proportion of baby weigh more than $13$ poundsis $28.77\text{\%}$ .

Explanation of Solution

Given information:The mean is $11.5$ and standard of deviation is $2.7$ .

Formula used:

  $Z=\frac{x-\mu }{\sigma }$

Here, the mean is $\mu$ and standard of deviation is $\sigma$ .

Calculation:

Consider the formula for Z value.

  $Z=\frac{x-\mu }{\sigma }$ …….. (1)

Substitute $13$ for $x$ , $11.5$ for $\mu$ and $2.7$ for $\sigma$ in Equation (1).

  $\begin{array}{c}Z=\frac{13-11.5}{2.7}\\ =0.56\end{array}$

From the table of standard normal distribution, for the Z value less than $0.56$ the value of proportion is $\text{0}\text{.71226}$ .

The proportion of baby weighs more than $13$ poundsis,

  $P\left(X>13\right)=1-P\left(Z<13\right)$ …….. (2)

Substitute the values in above equation.

  $\begin{array}{c}P\left(X>13\right)=1-\text{0}\text{.71226}\\ =0.28774\\ =28.77\text{\%}\end{array}$

Therefore, the proportion of baby weigh more than $13$ poundsis $28.77\text{\%}$ .

To determine

To find:The proportion of baby weighs less than $15$ pounds.

Answer to Problem 28E

The proportion of baby weighs less than $15$ poundsis $90.32\text{\%}$ .

Explanation of Solution

Given information:The mean is $11.5$ and standard of deviation is $2.7$ .

Formula used:

  $Z=\frac{x-\mu }{\sigma }$

Here, the mean is $\mu$ and standard of deviation is $\sigma$ .

Calculation:

Consider the formula for Z value.

  $Z=\frac{x-\mu }{\sigma }$ …….. (1)

Substitute $15$ for $x$ , $11.5$ for $\mu$ and $2.7$ for $\sigma$ in Equation (1).

  $\begin{array}{c}Z=\frac{15-11.5}{2.7}\\ =1.30\end{array}$

From the table of standard normal distribution, for the Z value less than $1.30$ the value of proportion is $\text{0}\text{.90320}$ .

Therefore, the proportion of baby weighs less than $15$ pounds is $90.32\text{\%}$ .

To determine

To find: The proportion of baby weighs between $10$ and $14$ pounds.

Answer to Problem 28E

The proportion of baby weighs between $10$ and $14$ poundsis $53.61\text{\%}$ .

Explanation of Solution

Given information:The mean is $11.5$ and standard of deviation is $2.7$ .

Formula used:

  $Z=\frac{x-\mu }{\sigma }$

Here, the mean is $\mu$ and standard of deviation is $\sigma$ .

Calculation:

Consider the formula for Z value.

  $Z=\frac{x-\mu }{\sigma }$ …….. (1)

Substitute $14$ for $x$ , $11.5$ for $\mu$ and $2.7$ for $\sigma$ in Equation (1).

  $\begin{array}{c}Z=\frac{14-11.5}{2.7}\\ =0.93\end{array}$

From the table of standard normal distribution, for the Z value less than $0.93$ the value of proportion is $\text{0}\text{.82381}$ .

Substitute $10$ for $x$ , $11.5$ for $\mu$ and $2.7$ for $\sigma$ in Equation (1).

  $\begin{array}{c}Z=\frac{10-11.5}{2.7}\\ =-0.56\end{array}$

From the table of standard normal distribution, for the Z value greater than $-0.56$ the value of proportion is $\text{0}\text{.28774}$ .

The proportion of baby weighs between $10$ and $14$ poundsis,

  $P\left(10<X<14\right)=P\left(Z<14\right)-P\left(Z>10\right)$ …….. (2)

Substitute the values in above equation.

  $\begin{array}{c}P\left(10<X<14\right)=0.82381-0.28774\\ =0.53607\\ =53.61\text{\%}\end{array}$

Therefore, the proportion of baby weighs between $10$ and $14$ poundsis $16.85\text{\%}$ .

To determine

To find:Whether it is unusual for a baby to weigh morethan $17$ .

Answer to Problem 28E

It is unusual for a baby to weigh more than $17$ .

Explanation of Solution

Given information:The mean is $11.5$ and standard of deviation is $2.7$ .

Calculation:

The point value of mean is,

  $\begin{array}{c}\mu \pm 2\sigma =11.5\pm 2\left(2.7\right)\\ =6.1,16.9\end{array}$ …….. (1)

Since, the value do not lies in between the interval.

Therefore, it is unusual for a baby to weigh more than $17$ .