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a set of data whose histogram is bell shaped yields a mean and standard deviation of 50 and 4, respectively, approximately what proportion of observation

a. between 46 and 54?

Respuesta :

LammettHash
[tex]\mathbb P(46<X<54)=\mathbb P\left(\dfrac{46-50}4<\dfrac{X-50}4<\dfrac{54-50}4\right)=\mathbb P(-1<Z<1)[/tex]

The empirical rule says that approximately 68% of the data lies within one standard deviation of the mean.

The proportion of observation that lies between [tex]46[/tex] and [tex]54[/tex] is [tex]68\%[/tex].

What is empirical rule ?

Empirical rule : In statistics, the [tex]68-95-99.7[/tex] rule, also known as the empirical rule, is a shorthand used to remember the percentage of values that lie within an interval estimate in a normal distribution: [tex]68\%,[/tex] of the values lie within one standard deviations of the mean.

i.e. [tex]\mu -\sigma[/tex] and [tex]\mu+\sigma[/tex]

We have,

Mean [tex](\mu)=50[/tex]

Standard deviation [tex](\sigma)=50[/tex]

So,

Now Using the Empirical rule,

[tex]\mu -\sigma[/tex] and [tex]\mu+\sigma[/tex]

i.e.

[tex]50-4=46[/tex]

and, [tex]50+4=54[/tex]

That means,

That data lie between  [tex]46[/tex] and [tex]54[/tex] is [tex]68\%[/tex].

Hence, we can say that the proportion of observation that lies between [tex]46[/tex] and [tex]54[/tex] is [tex]68\%[/tex].

To know more about Empirical rule click here

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