Suppose a sample of 510 new car buyers is drawn. Of those sampled, 142 preferred foreign over domestic cars. Using the data, construct the 85% confidence interval for the population proportion of new car buyers who prefer foreign cars over domestic cars. Round your answers to three decimal places.

(0.224, 0.332)

Step-by-step explanation: from the question, the total population is 510 and 142 out of this population prefers foreign cars to domestic ones.

Hence the sample size (n) is 142.

Sample proportion (p) = 142/ 510 = 0.278

q = 1 - 0.278 = 0.722.

We are to construct a 85% confidence interval for sample proportion and this is given by the formulae below.

P = p + Zα/2×(√(pq/n) For upper limit

P = p - Zα/2×(√(pq/n) For lower limit

We are using a z test to get our critical value because sample size is greater than 30 ( n = 142).

The value of Zα/2 from the standard normal distribution table is 1.44 ( this is done at a 15% level of significance).

By substituting the parameters, we have that

For upper limit

P = 0.278 + 1.44 × (√(0.278×0.722/142)

P = 0.278 + 1.44(0.0375)

P = 0.278 + 0.054

P = 0.332

For lower limit

P = 0.278 - 1.44 × (√(0.278×0.722/142)

P = 0.278 - 1.44(0.0375)

P = 0.278 - 0.054

P = 0.224.

Hence the 85% confidence interval for population proportion is (0.224, 0.332)

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!

- "runner b is moving at a slower rate."

- "the graphs shown are graphs of distance(km) against time(min).

the slopes of this graphs are are given by:

slope = (change in y)/(change in x)

= (change in distance)/(change in time)

this is the formula for finding the speed. so, the slope of these graphs represent the speed of the runner.

taking each division in the graph to represent 1 on both axis:

runner a ⇒ speed = (2-0)/(1-0)

= 2/1

= 2 km/min

runner b ⇒ speed = (3-0)/(3-0)

= 3/3

= 1 km/min

this shows that   runner b is moving at a slower rate."