Question

Intensity Frequency 1 450 2 510 3 213 4 98 5 46 1317 Total Find the mean intensity number: Oa 2 Ob. 2.07 Oc. 3 Od. 546.20 19. If we have a population of test scores with u = 80 and o=6 that is bell-shaped, use the Empirical Rule to derive an interval in which about 95% of the scores fall.:* Oa (74, 86) Ob. (68, 92) Oc. (60, 98) Od. (68, 86) 20. The statistic that describes the fiftieth percentile in a sample is known as what?:* Oa. Mean Ob. Standard deviation Oc. Upper quartile Od. Median

Answer 1

Question 18) -

We find the mean number by using the following formula,

ean Στ.fi Σ.fi

$x_i$ is the i^th intensity and is the frequency if i^th intensity

fi

Intensity ($x_i$)	Frequency ( fi )	$x_i$ fi
1	450	450
2	510	1020
3	213	639
4	98	392
5	46	230
Total	1317	2731

Mean = 2731 / 1317 = 2.0736

Hence, correct option for mean intensity number is 2.07

Question 19) -

Let X denote the test score

It is given that X follows bell shaped (normal) distribution with mean $\mu$ = 80 and standard deviation $\sigma$ = 6

$Z=\frac{X-\mu}{\sigma}$ follows standard normal distribution.

We need to find interval (a,b) such that P[ a<X<b ] = 0.95

$P[\frac{a-\mu}{\sigma}<\frac{X-\mu}{\sigma}<\frac{b-\mu}{\sigma}]=0.95$

Probability outside the interval is 0.05. Hence, probability below lower limit is 0.025 and above upper limit is 0.025.

That is, $P[\frac{X-\mu}{\sigma}<\frac{b-\mu}{\sigma}]=0.975$ and $P[\frac{X-\mu}{\sigma}<\frac{a-\mu}{\sigma}]=0.025$

From the standard normal probability tables we have that,

P[ Z < -1.96 ] = 0.025 and P[ Z < 1.96 ] = 0.975

Hence, $\frac{a-\mu}{\sigma}=-1.96\Rightarrow a=\mu-1.96\sigma=68$

$\frac{b-\mu}{\sigma}=1.96\Rightarrow b=\mu+1.96\sigma=92$

Hence, an interval in which 95% of the scores fall is ( 68, 92 )

Question 20) -

According to the definition Median M is defined as the point below which 50% of the observations lies and above which rest of the 50% of the observations lie.

That is, P[ X < M ] = P[ X > M ] = 50% = 0.5

Hence, The statistic that describes the fiftieth percentile in a sample is known as Median.

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