Question

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?

Answer 1

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.