1. Given a regular octagon, find the measures of the angles formed by (a) two consecutive radii and (b) a radius and a side of the polygon. a. 40°; 230° b. 45°; 67.5° c. 72°; 252° d. 36°; 216°

(B)45°, 67.5°

Step-by-step explanation:

A regular polygon is a polygon in which all the sides are equal.

An Octagon is a polygon with 8 sides.

Since it is a Regular polygon, the following holds:

1.The Radii are equal

2. The angle Subtended by two radii are equal for all triangles in the octagon.

3. There are 8 Isosceles triangles in the octagon.

(a)Measures of angle formed by Two radii

From the centre of the octagon, there are 8 equal angles.

Sum of Angle at a point= 360°

Each Angle=360°÷ 8 = 45°

The measures of angle formed by two radii is 45°.

(b)a radius and a side of the polygon.

Since the triangles are Isosceles, the base angles at the side of the octagon are equal.

Let x be the angle made by a radius with a side.

From (a) above,

x+x+45°=180°(Sum of Angle in a Triangle)

2x=180-45°

2x=135°

x=135°÷2=67.5°

The angle between a radius and a side of the polygon is 67.5°.

a) Null hypothesis:  

Alternative hypothesis:  

b)  

c) If we compare the p value and the significance level given we see that so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean for the salary differs from 9000 at 5% of significance.

Step-by-step explanation:

1) Data given and notation  

77600 ,76000 ,90700 ,97200 ,90700 ,101800 ,78700 ,81300 ,84200 ,97600 ,

77500 ,75700 ,89400 ,84300 ,78700 ,84600 ,87700 ,103400 ,83800 ,101300

94700 ,69200 ,95400 ,61500 ,68800

We can calculate the sample mean and deviation with the following formulas:

The values obtained are:

represent the mean annual salary for the sample  

represent the sample standard deviation for the sample  

sample size  

represent the value that we want to test

represent the significance level for the hypothesis test.  

t would represent the statistic (variable of interest)  

represent the p value for the test (variable of interest)  

Part a: State the null and alternative hypotheses.  

We need to conduct a hypothesis in order to check if the mean salary differs from 90000, the system of hypothesis would be:  

Null hypothesis:  

Alternative hypothesis:  

If we analyze the size for the sample is < 30 and we don't know the population deviation so is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:  

 (1)  

t-test: "Is used to compare group means. Is one of the most common tests and is used to determine if the mean is (higher, less or not equal) to an specified value".  

Part b: Calculate the statistic

We can replace in formula (1) the info given like this:  

P-value

The first step is calculate the degrees of freedom, on this case:  

Since is a two sided test the p value would be:  

Part c: Conclusion  

If we compare the p value and the significance level given we see that so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean for the salary differs from 9000 at 5% of significance.