G In a certain rural county, a public health researcher spoke with 111 residents 65-years or older, and 28 of them had obtained a flu shot. The researcher wants to calculate a 95% confidence interval for the percent of the 65-plus population that were getting the flu shot. Historically in this county, 30% of residents typically obtain a flu shot each year.

95% confidence interval for the percent of the 65-plus population that were getting the flu shot is [0.169 , 0.331].

Step-by-step explanation:

We are given that in a certain rural county, a public health researcher spoke with 111 residents 65-years or older, and 28 of them had obtained a flu shot.

Firstly, the Pivotal quantity for 95% confidence interval for the population proportion is given by;

                          P.Q. =    ~ N(0,1)

where, = sample proportion of residents 65-years or older who had obtained a flu shot = = 0.25

          n = sample of residents 65-years or older = 111

          p = population proportion of residents who were getting the flu shot

Here for constructing 95% confidence interval we have used One-sample z test for proportions.

So, 95% confidence interval for the population proportion, p is ;

P(-1.96 < N(0,1) < 1.96) = 0.95  {As the critical value of z at 2.5% level

                                                of significance are -1.96 & 1.96}  

P(-1.96 < < 1.96) = 0.95

P( < < ) = 0.95

P( < p < ) = 0.95

95% confidence interval for p = [ , ]

   = [ , ]

   = [0.169 , 0.331]

Therefore, 95% confidence interval for the percent of the 65-plus population that were getting the flu shot is [0.169 , 0.331].

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