Answer:
The 98% confidence interval
[tex]0.1884 < p < 0.2876[/tex]
The confidence interval contains 20.4%
Step-by-step explanation:
From the question we are told that
The sample size is n = 400
The number that replied that they were considering business as a major [tex]x = 95[/tex]
The sample proportion is mathematically evaluated as
[tex]\r p = \frac{95}{400}[/tex]
[tex]\r p = 0.238[/tex]
Given that the confidence level 98% then the level of significance is evaluated as
[tex]\alpha = 100 - 98[/tex]
[tex]\alpha = 2 \%[/tex]
[tex]\alpha = 0.02[/tex]
Next we obtain the critical value of [tex]\frac{ \alpha }{2}[/tex] from the normal distribution table is
[tex]Z_{\frac{ \alpha }{2} } = 2.33[/tex]
Generally the margin of error is mathematically represented as
[tex]E = Z_{\frac{ \alpha }{2} } * \sqrt{ \frac{ p (1 - p )}{n} }[/tex]
[tex]E = 2.33 * \sqrt{ \frac{ 0.238 (1 - 0.238 )}{400} }[/tex]
[tex]E = 0.0496[/tex]
The 98% confidence interval is mathematically represented
[tex]\r p - E < p < \r p + E[/tex]
=> [tex]0.238 - 0.0496 < p <0.238 + 0.0496[/tex]
=> [tex]0.1884 < p < 0.2876[/tex]
