## IB Maths HL Option Statistics Probability Normal

Expectation of product of independent random
variables, General unbiased estimators, efficiency
of estimators, Bivariate distributions, Covariance, Pearson’s product moment correlation coefficient, Probability generating functions, Cumulative distribution function now explicitly
for both discrete and continuous distributions,unbiased estimators

### IB Maths HL Option Statistics Probability Normal

IB Mathematics HL – Option Statistics Probability Normal Distribution Continuous Probability Distribution

How can we find the standard deviation of the weight of a population of dogs which is found to be normally distributed with mean 7.4 Kg and the 30% of the dogs weigh at least 8 Kg.

Thanks
elizabeth

Posts: 0
Joined: Mon Jan 28, 2013 8:09 pm

### Re: IB Maths HL Option Statistics Probability Normal

IB Mathematics HL – Option Statistics Probability Normal Distribution Continuous Probability Distribution

A normal distribution is a continuous probability distribution for a random variable X. The graph of a normal distribution is called the normal curve. A normal distribution has the following properties.
1. The mean, median, and mode are equal.
2. The normal curve is bell shaped and is symmetric about the mean.
3. The total are under the normal curve is equal to one.
4. The normal curve approaches, but never touches, the x-axis as it extends farther and farther away from the mean.

Approximately 68% of the area under the normal curve is between $\mu - \sigma$ and $\mu + \sigma$
and . Approximately 95% of the area under the normal curve is between $\mu-2 \sigma$ and $\mu+2 \sigma$. Approximately 99.7% of the area under the normal curve is between $\mu-3 \sigma$ and $\mu+3 \sigma$

The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Let the random variable W denote the weight of the dogs, so that

$W\sim N(7.4, \sigma ^2)$

We know that $P(W \geq 8)=0.3$

Since we don’t know the standard deviation, we cannot use the inverse normal. Therefore we have to transform the random variable $W$ to that of

$Z\sim N(0, 1)$ , using the transformation $Z= \frac{X- \mu}{\sigma}$

we have the following

$P(W \geq 8)=0.3 \Rightarrow P(\frac{X- \mu}{\sigma} \geq \frac{8- 7.4}{\sigma})=0.3$

$\Rightarrow P(Z \geq \frac{0.6}{\sigma})=0.3$

Using GDC Casio fx-9860G SD
MAIN MENU > STAT>DIST(F5)>NORM(F1)>InvN>

Setting Tail: right
Area: 0.3
$\sigma$:1
$\mu$:0

We find that the standardized value is 0.5244

Therefore
$\frac{0.6}{\sigma}=0.5244 \Rightarrow \sigma=\frac{0.6}{0.5244}=1.14 (3 s.f.)$

Hope these help!!
miranda

Posts: 268
Joined: Mon Jan 28, 2013 8:03 pm

### Re: IB Maths HL Option Statistics Probability Normal

Thank you miranda!!!
elizabeth

Posts: 0
Joined: Mon Jan 28, 2013 8:09 pm

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