Observed proportions (relative frequencies) from a sample are statistics that can be used to estimate probabilities.
a) Thre is 99.73% proportion of observations is in between 190 and 310 .
b)there is 95.44% proportion of observations is in between 210 and 290 .
We have given that,
Mean of population, μ = 250
standard deviations, σ = 20
we have to calculate proportion of observations in the given interval pair.
a) pair of values are 190 and 310
Using Z-score formula,
Z =( x - μ) /σ
=> Z = (190 - 250) / 20 = -3
Similarly, Z = (310 - 250)/20 = 3
So, -3 < z < 3 and from Z score we found p-value
P(190<X<310) = P( 190-u/sd < X-u/sd < 310-u/sd)
= P( X< 310 ) - P(X<190) = P(Z < 3) - P( Z< -3 )
=0.99865 - 0.00135 0.9973 = 99.73%
b) pair is 210 and 290
Z-Score = (210-250)/20 = -2
Z-score = (290-250)/20 = 2
so, -2 < z < 2
From Z score we found p-value
P(210<X<290) = P(X<290) - P(X<210)
= P(Z< 2) - P(Z< -2)
= 0.9772 - 0.0228 = 0.9544 = 95.44%
So, the required proportion are 99.73% and 95.44% proportion of observations in each given pair.
To learn more about Proportion value , refer:
https://brainly.com/question/19131394
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Complete question:
If the mean of a population is 250 and its standard deviation is 20, approximately what proportion of observations is in the interval between each pair of values? a 190 and 310 b. 210 and 290.
