Question

A researcher reported that 71.8 of all email sent in a recent month was spam. A system manager at a large corporation believes that the percentage at his company may be 76. He examines a random sample of emails received at an email server, and finds that 360 of the messages are spam. Can you conclude that the percentage of emails that are spam differs from 76

Answer 1

Following are the solution to the given question:

Explanation:

`n = 500\\\\X = 360\\\\\hat{p}=(X)/(n)\\\\`

`=(360)/(500)\\\\=(36)/(50)\\\\= 0.72`

Calculating the Null hypothesis:

`H_(0):p=0.76`

Calculating the Alternative hypothesis:

`H_(1):p\\eq 0.76`

`\therefore \\\\p = 0.76\\\\\because \\\\ q = 1 - p = 1 - 0.76 = 0.24\\\\Test \ \ statistics\ Z = \frac{(\hat{p}-p)}{\sqrt{(pq)/(n)}}\\\\`

`= \frac{(0.72-0.76)}{\sqrt{((0.76* 0.24))/(500)}}\\\\= -\frac{0.04}{\sqrt{(0.1824)/(500)}}\\\\= -(0.04)/(√(0.000365))\\\\= -(0.04)/(0.019)\\\\= -2.10`

`\alpha=0.05\\\\p-value = 0.0179\\\\p-value< 0.05\\\\\therefore \\\\0.0179 < 0.05, \text{reject null hypothesis}\\\\P \leq 0.05,\text{reject null hypothesis} \ H_(0)\ at\ \alpha=0.05 \\\\\alpha=0.10\\\\ p-value = 0.0179\\\\p-value< 0.10\\\\\therefore \ 0.0179 < 0.10, \text{reject null hypothesis}\\\\P > 0.10,\text{we do not reject null hypothesis}\ H_(0) \ at \alpha=0.10 .`