Apostmix beverage machine is adjusted to release a certain amount of syrup into a chamber where it is mixed with carbonated water. a random sample of 25 beverages was found to have a mean syrup content of x¯=1.19 fluid ounces and the sample standard deviation is s=0.015 fluid ounces. find a 95% two-sided confidence interval on the mean volume of syrup dispensed. assume population is approximately normally distributed.

  (1.1838, 1.1962)

Step-by-step explanation:

The formula we use to calculate the confidence interval for population mean ( if population standard deviation is not given)is given by :-

,

where n= sample size

s= sample standard deviation.

= sample mean

t* = Two-tailed critical t-value.

Given : n= 25

Degree of freedom : df = n-1 =24

Significance level

Now from students' t-distribution table , check the t-value for significance level and df=24:

t*=2.0639

 fluid ounces

s= 0.015  fluid ounces

We assume that the population is approximately normally distributed

Now, the 95% two-sided confidence interval on the mean volume of syrup dispensed :-

∴ The required confidence interval = (1.1838, 1.1962)

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step-by-step explanation:

the value of x is 9 ⇒ answer b

step-by-step explanation:

* we can solve this problem using cosine and sine rule

- in δ abc

∵ ab = 36 , bc = 28 , ac = 16

- lets find the measure of angle b using the cosine rule

∵ cos b = [ab² + bc² - ac²]/2(ab)(bc)

∴ cos b = [36² + 28² - 16²]/2(36)(28)

∴ cos b = 19/21

∴ m∠b = cos^-1(19/21) ≅ 25°

- lets find the measure of angle a using the sine rule

∵ sina/bc = sinb/ac = sinc/ab

∴ sina/28 = sin25°/16 ⇒ by using cross multiplication

∴ sina = 28 × sin25°/16

∴ sina = 0.7396

∴ m∠a = sin^-1(0.7396) ≅ 48°

- from the figure m∠abd = m∠dbc

∵ m∠abc = 25°

∴ m∠abd = 25°/2 = 12.5°

- in δabd

∵ m∠a = 48° , m∠abd = 12.5°

∵ the sum of the measures of the interior angles of any δ is 180°

∴ m∠adb = 180° - (48° + 12.5°) = 119.5°

∵ ab = 36

∵ ad = x

- use the sine rule to find x

∵ sin∠adb/ab = sin∠abd/x

∴ sin119.5°/36 = sin12.5°/x ⇒ by using cross multiplication

∴ x = 36 × sin12.5°/sin119.5° = 8.95 ≅ 9

* the value of x is 9