Complete Question
The complete question is shown on the first uploaded image
Answer:
The value is [tex]n = 146093[/tex]
Step-by-step explanation:
From the question we are told that
The margin of error is [tex]E = 0.04[/tex]
The data is
8.70 4.79 10.95 15.19 14.06
16.99 1.22 9.02 14.39 5.73
7.28 3.22 2.66 6.13 6.93
Generally the sample mean is mathematically represented as
[tex]\= x = \frac{\sum x_i}{n}[/tex]
=> [tex]\= x = \frac{ 8.70 + 4.79 +\cdots + 6.93}{15}[/tex]
=> [tex]\= x = 8.48[/tex]
Generally the standard deviation is mathematically represented as
[tex]\sigma = \sqrt{ \frac{ \sum (x_i - \= x)}{n} }[/tex]
=> [tex]\sigma = \sqrt{ \frac{ (8.70 - 8.48 )^2 + ( 8.70 - 8.48)^2 + \cdots + (6.93 - 8.48)^2 }{15} }[/tex]
=> [tex]\sigma = 5.14[/tex]
From the question we are told the confidence level is 98% , hence the level of significance is
[tex]\alpha = (100 - 98 ) \%[/tex]
=> [tex]\alpha = 0.02[/tex]
Generally the degree of freedom is mathematically represented as
[tex]df = n -1[/tex]
=> [tex]df = 15 -1[/tex]
=> [tex]df = 14[/tex]
Generally from the t distribution table the critical value of at a degree of freedom of is
[tex]t_{\frac{\alpha }{2} , 14 } = 2.9768 [/tex]
Generally the sample size is mathematically represented as
[tex]n =[ \frac{t_{\frac{\alpha }{2} , 14 } * \sigma }{E} ]^2[/tex]
=> [tex]n =[ \frac{ 2.9768 * 5.136 }{0.04} ]^2[/tex]
=> [tex]n = 146093[/tex]
