Question

problem 2
the weight of a population of seals is approximately normally distributed with a mean of 80 kg and a standard deviation of 8 kg. using technology or a standard normal table, estimate the proportion (as a decimal and percentage) of seals in each category.

1. what proportion of seals weigh less than 88 kg?
2. what proportion of seals weigh between 72 kg and 96 kg?
3. what proportion of seals weigh more than 96 kg?

Explanation:

Step1: Calculate z-score for 88 kg

The z-score formula is $z=\frac{x-\mu}{\sigma}$, where $\mu=80$, $\sigma=8$, $x=88$.
$z=\frac{88-80}{8}=1$

Step2: Find proportion for z=1

Using standard normal table, $P(Z<1)=0.8413$

Step3: Calculate z-scores for 72 kg and 96 kg

For $x=72$: $z=\frac{72-80}{8}=-1$
For $x=96$: $z=\frac{96-80}{8}=2$

Step4: Find proportion between z=-1 and z=2

$P(-1

Step5: Find proportion for z=2 (weight >96 kg)

$P(Z>2)=1-P(Z<2)=1-0.9772=0.0228$

Step6: Convert decimals to percentages

Multiply each decimal by 100.

Answer:

1. Decimal: $0.8413$, Percentage: $84.13\%$
2. Decimal: $0.8185$, Percentage: $81.85\%$
3. Decimal: $0.0228$, Percentage: $2.28\%$