Write as a percent. (if necessary, round to the nearest tenth of a percent.)13/30

A simple random sample of size n is drawn from a population that is normally distributed. The sample mean, x overbar, is found

joja [24]

Answer:

(a) 80% confidence interval for the population mean is [109.24 , 116.76].

(b) 80% confidence interval for the population mean is [109.86 , 116.14].

(c) 98% confidence interval for the population mean is [105.56 , 120.44].

(d) No, we could not have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed.

Step-by-step explanation:

We are given that a simple random sample of size n is drawn from a population that is normally distributed.

The sample mean is found to be 113 and the sample standard deviation is found to be 10.

(a) The sample size given is n = 13.

Firstly, the pivotal quantity for 80% confidence interval for the population mean is given by;

P.Q. = $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ ~ $t_n_-_1$

where, $\bar X$ = sample mean = 113

s = sample standard deviation = 10

n = sample size = 13

$\mu$ = population mean

Here for constructing 80% confidence interval we have used One-sample t test statistics as we don't know about population standard deviation.

So, 80% confidence interval for the population mean, $\mu$ is ;

P(-1.356 < $t_1_2$ < 1.356) = 0.80 {As the critical value of t at 12 degree

of freedom are -1.356 & 1.356 with P = 10%}

P(-1.356 < $\frac{\bar X-\mu}{\frac{s}{\sqrt{n} } }$ < 1.356) = 0.80

P( $-1.356 \times }{\frac{s}{\sqrt{n} } }$ < ${\bar X-\mu}$ < $1.356 \times }{\frac{s}{\sqrt{n} } }$ ) = 0.80

P( $\bar X-1.356 \times }{\frac{s}{\sqrt{n} } }$ < $\mu$ < $\bar X+1.356 \times }{\frac{s}{\sqrt{n} } }$ ) = 0.80

80% confidence interval for $\mu$ = [ $\bar X-1.356 \times }{\frac{s}{\sqrt{n} } }$ , $\bar X+1.356 \times }{\frac{s}{\sqrt{n} } }$ ]

= [ $113-1.356 \times }{\frac{10}{\sqrt{13} } }$ , $113+1.356 \times }{\frac{10}{\sqrt{13} } }$ ]

= [109.24 , 116.76]

Therefore, 80% confidence interval for the population mean is [109.24 , 116.76].

(b) Now, the sample size has been changed to 18, i.e; n = 18.

So, the critical values of t at 17 degree of freedom would now be -1.333 & 1.333 with P = 10%.

80% confidence interval for $\mu$ = [ $\bar X-1.333 \times }{\frac{s}{\sqrt{n} } }$ , $\bar X+1.333 \times }{\frac{s}{\sqrt{n} } }$ ]

= [ $113-1.333 \times }{\frac{10}{\sqrt{18} } }$ , $113+1.333 \times }{\frac{10}{\sqrt{18} } }$ ]

= [109.86 , 116.14]

Therefore, 80% confidence interval for the population mean is [109.86 , 116.14].

(c) Now, we have to construct 98% confidence interval with sample size, n = 13.

So, the critical values of t at 12 degree of freedom would now be -2.681 & 2.681 with P = 1%.

98% confidence interval for $\mu$ = [ $\bar X-2.681 \times }{\frac{s}{\sqrt{n} } }$ , $\bar X+2.681 \times }{\frac{s}{\sqrt{n} } }$ ]

= [ $113-2.681 \times }{\frac{10}{\sqrt{13} } }$ , $113+2.681 \times }{\frac{10}{\sqrt{13} } }$ ]

= [105.56 , 120.44]

Therefore, 98% confidence interval for the population mean is [105.56 , 120.44].

(d) No, we could not have computed the confidence intervals in parts (a)-(c) if the population had not been normally distributed because t test statistics is used only when the data follows normal distribution.

6 0

3 years ago

1. Given f(x)=2x 2 +1 and g(x)=3x-5 find the following

LenKa [72]

The value of the function 1) f -g = 2x² - 3x + 6 and 2) f(g(2)) is 3.

Here the two functions are given, f(x) and g(x).

f(x) = 2x² + 1

g(x) = 3x - 5

We have to find f-g and f(g(2)).

1) f- g

f(x) - g(x)

(2x² + 1) - ( 3x - 5)

2x² + 1 - 3x + 5

2x² - 3x + 6

2) f(g(2))

f(g(x)) = 2(3x-5)² + 1

= 2( 9x² - 30x + 25) + 1

= 18x² - 60x + 50 + 1

= 18x² - 60x + 51

f(g(2)) = 18(2)² - 60(2) + 51

=18× 4 - 120 + 51

= 72 - 120 + 51

= 123 - 120

= 3

Therefore the value of f-g is 2x²- 3x + 6 and the value of f(g(2)) is 3.

To know more about the function refer to the link given below:

brainly.com/question/11624077

#SPJ4

4 0

1 year ago

Truck or treat logic grid I need help plz The picture is the clues

erastova [34]

Answer: Can you make the question a bit clear?

4 0

3 years ago

Magdalena bought a sweater that costs “d” dollars she paid the clerk $40 she received $5.19 in change which equation represents

trasher [3.6K]

Answer:

40-d=5.19

Step-by-step explanation:

3 0

3 years ago

Read 2 more answers

2 3/5 ÷ 2/7 9 1/10 26/35 6/35 4 1/10 choose one please

Step2247 [10]

Answer:

The answer is 91/10

Step-by-step explanation:

hope this helps

Solutions:

2 3/5÷2/7

13/5÷2/7

13/5×7/2

=13×7=91

=5×2=10

That's why the answer is 91/10

6 0

3 years ago