If the distribution is really N ( 5.43 , 0.54 ) , N(5.43,0.54), what proportion of observations would be less than 5.05 ? 5.05? (Enter your answer rounded to four decimal places.)

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".

Solution to the problem

Let X the random variable of interest for a population, and for this case we know the distribution for X is given by:

[tex]X \sim N(5.43,0.54)[/tex]

Where [tex]\mu=5.43[/tex] and [tex]\sigma=0.54[/tex]

We are interested on this probability

[tex]P(X<5.05)[/tex]

And the best way to solve this problem is using the normal standard distribution and the z score given by:

[tex]z=\frac{x-\mu}{\sigma}[/tex]

If we apply this formula to our probability we got this:

[tex]P(X<5.05)=P(\frac{X-\mu}{\sigma}<\frac{5.05-\mu}{\sigma})=P(Z<\frac{5.05-5.43}{0.54})=P(Z<-0.704)[/tex]

And we can find this probability on this way:

[tex]P(Z<-0.704)=0.241[/tex]

And we can use the following excel code to find the answer: "=NORM.DIST(-0.704,0,1,TRUE)"