The following sample data are from a normal population: 12 7 6 3 5 12 9 10 11 2 Calculate the point estimate of the population mean.

The sample mean (x⁻ ) is the point of estimate of the Population mean

x⁻ = 7.7

The sample standard deviation   (S)  is the point of estimate of the Population standard deviation

 S = 3.65

Step-by-step explanation:

Step(i):-

Given data    12  7  6  3   5   12   9   10  11  2

Mean of the data

x          :     12      7       6       3        5         12      9       10           11        2

x-x⁻      :   4.3     -0.7    -1.7   -4.7    -2.7     4.3    1.3      2.3         3.3     -5.7

(x-x⁻  )² :  18.49  0.49   2.89  22.09 7.29  18.49 1.69    5.29  10.89   32.49

Step(ii):-

Standard deviation of the sample

Standard deviation of the sample

 S = √13.344 =3.65

Final answer:-

The sample mean (x⁻ ) is the point of estimate of the Population mean

x⁻ = 7.7

The sample standard deviation   (S)  is the point of estimate of the Population standard deviation

 S = 3.65

∠1≅∠2 by the alternate exterior angles theorem.

step-by-step explanation:

given, a ∥ b and ∠1 ≅ ∠3 .we have to prove that e ∥ f

we know that ∠1≅∠3 and that a || b because they are given. we see that by the alternate exterior angles theorem. therefore, ∠2≅∠3 by the transitive property. so, we can conclude that e || f by the converse alternate exterior angles theorem.

we have to fill the missing statement.

transitivity property states that if a = b and b = c, then a = c.

now, given ∠1≅∠3 and by transitivity property ∠2≅∠3 .

hence, to apply transitivity property one angle must be common which is not in result after applying this property which is ∠1.

the only options in which ∠1 is present are ∠1 and ∠2, ∠1 and ∠4

∠1 and ∠4 is not possible ∵ after applying transitivity we didn't get ∠4.

hence, the missing statement is ∠1≅∠2.

so, ∠1≅∠2 by the alternate exterior angles theorem.