A researcher reported that 71.8% of all email sent in a recent month was spam. A system large corporation believes that the percentage at his company may be 77 % . He manager at a examines a random sample of 500 emails received at an email server, and finds that 361 of the messages are spam. Required:
Can you conclude that less than 77% of emails are spam?

p-value = 0.0054

at 0.05 level  of significance ∝

p-value ( 0.0054 ) is less than significance level ∝ ( 0.05 )

Thus we Reject Null hypothesis  H₀

Hence, there is enough evidence to conclude that less than 77% of emails are spam.

Step-by-step explanation:

Given the data in the question;

Sample size n = 500

x = 361

p₀ = 77% = 0.77

q₀ = 1 - p₀ = 1 - 0.77 = 0.23

p = x / n = 361 / 500 = 0.722

Hypothesis;

Null hypothesis             H₀ : p ≥  77% or 0.77

Alternative hypothesis Hₐ : p ≥  77% or 0.77

Test Statistic

z = ( p - p₀ ) / √( ( p₀ × q₀ ) / n )  

so we substitute

z = ( 0.722 - 0.77 ) / √( ( 0.77 × 0.23 ) / 500 )  

z = -0.048 / √( 0.1771 / 500 )  

z = -0.048 / √0.0003542  

z = -0.048 / 0.01882

z = -2.55

so our p-value will be;

p-value = p( z < -2.55 ) {from table }

p-value = 0.0054

at 0.05 level  of significance ∝

p-value ( 0.0054 ) is less than significance level ∝ ( 0.05 )

Thus we Reject Null hypothesis  H₀

Hence, there is enough evidence to conclude that less than 77% of emails are spam.

something that characterises a right triangle is the fact that we may use pythagoras' theorem to find any one of its sides given that we have the length of the other two sides. here, we have the length of the hypotenuse, √(87) yd, and one other side, √(23) yd, thus we can easily find the length of the remaining side using pythagoras' theorem as defined below:

c^2 = a^2 + b^2, where c is the length of the hypotenuse, and a and b are the lengths of the other two sides.

thus, given that c = √(87) and a = √(23), we may substitute these into the formula we defined above to get:

c^2 = a^2 + b^2

(√87)^2 = (√23)^2 + b^2 (substitute c = √87 and a = √23 into the equation)

87 = 23 + b^2 (evaluate (√87)^2 = 87 and (√23)^2 = 23 - remember that the square root of a number, squared will equal that very same number)

64 = b^2 (subtract 23 from both sides)

√64 = b (take the square root of both sides)

8 = b (evaluate √64 = 8)

therefor, the length of the unknown leg in the right triangle is 8 yd (first answer).

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

step-by-step explanation:

given   that triangle lno is congruent to triangle lnm.

we have to fill up the blanks in the given question.

according to above order of congruency   kept in mind we can say that

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

ln ≅ ln due to common in both.

hence, by sas congruency,

the side angle side rule   states that if two sides and the included angle of one triangle are congruent to another triangle, then these two triangles are congruent.