Question

Assume that a set of test scores is normally distributed with a mean of 80 and a standard deviation of 15. Use the​ 68-95-99.7 rule to find the following quantities. a. The percentage of scores less than 80 is 50​%. ​(Round to one decimal place as​ needed.) b. The percentage of scores greater than 95 is 16.0​%. ​(Round to one decimal place as​ needed.) c. The percentage of scores between 50 and 95 is nothing​%.

Answer 1

Answer:

a. Percentage of scores less than 100::::: 50%

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b. Relative frequency of scores less than 120:::(50+34)%

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c. Percentage of scores less than 140:::::(50+47.5)%

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d. Percentage of scores less than 80:::::::(50-34)%

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e. Relative frequency of scores less than 60::::(50-47.5)%

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f. Percentage of scores greater than 120:::(50-34)%

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Step-by-step explanation:

Answer 2

The percentages will be "50.0%", "16.0%" and "81.5%". A further solution is provided below.

According to the question,

Mean,

• 80

Standard deviation,

• 15

(a)

The total area curve,

= 1.00

Half area under the curve,

= 0.500

then,

→ $P (X< 80) = 0.500$

or,

→                   $= 50.0$ (%)

(b)

As we know,

→ $Mean +1\times Standard \ deviation= 80+1\times 15$

                                                       $= 80+15$

                                                       $= 95$

According to 68-95-99.7 rule,

The remaining area of both sides will be:

= $100-68$

= $32$ (%)

hence,

Remaining area of right side will be:

→ $P(X> 95)$ = $\frac{32}{2}$

                    = $16$ (%)

(c)

We have,

• $P(X<95) = 0.16$

So,

→ $P(X< 95)=1-0.16$

                    $= 0.84$

Now,

→ $Mean-2\times Standard \ deviation= 80-2\times 15$

                                                       $=80-30$

                                                       $= 50$

According to 68-95-99.7 rule,

Remaining area of both sides will be:

= $1-.95$

= $0.05$

Area remaining at left,

= $\frac{0.05}{2}$

= $0.025$

So, $P(X<50) = 0.025$

hence,

→ $P(50<X<95) = P(X<95)-P(X<50)$

                             $= 0.84-0.025$

                             $= 0.815$

or,

                             $= 81.5$ (%)    

Thus the above answers are correct.

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