Two random samples of 40 students were drawn independently from two populations of students. assume their aptitude tests are normally distributed (total points = 100). the following statistics regarding their scores in an aptitude test were obtained: xbar1 = 76 s1= 8 xbar2 = 72 and s2= 6.5. do not assume population variances are equal. use all formulas assuming unequal variances.

test at the 5% significance level to determine whether we can infer that the two population means differ.

explain how to use the 95% confidence interval to test the hypotheses at a = .05.

Comparing the p value with a significance level for example we see that so we can conclude that we have enough evidence to reject the null hypothesis, and we can say that the population mean for group 1 is significantly different than the population mean for group 2.  

The 95% confidence interval is given by (0.756;7.243), since the interval not contain's the 0 we can reject the null hypothesis that the means are equal.

Step-by-step explanation:

Data given and notation

represent the mean for 1  

represent the mean for 2  

represent the sample standard deviation for 1  

represent the sample standard deviation for 2

sample size for the group 1  

sample size for the group 2  

Significance level provided  

t would represent the statistic (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to check if the population means differs, the system of  hypothesis would be:  

Null hypothesis:  

Alternative hypothesis:  

We don't have the population standard deviation's, we can apply a t test to compare means, 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 whether the means of two groups are equal to each other.  

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

P value  

We need to find first the degrees of freedom given by:

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

Comparing the p value with a significance level for example we see that so we can conclude that we have enough evidence to reject the null hypothesis, and we can say that the population mean for group 1 is significantly different than the population mean for group 2.  

Confidence interval

The critical value for this case can be calculated like this. and . The degrees of freedom are 78 so we can use this code in excel to find the critical value for the interval "=-T.INV(0.025,78)" and we got

The confidence interval for this case would be given by this formula:

And if we replace the values given we have:

The 95% confidence interval is given by (0.756;7.243), since the interval not contain's the 0 we can reject the null hypothesis that the means are equal.

a) $ 9.50

b) 17

c) y-intercept (0, 12). this is the initial amount of money that joe had prior to playing any game.

d) x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e) slope is -0.5. joe's balance will reduce by 0.5 for every game that he plays.

f)

domain; [0,24]

range; [0,12]

g) x + 2y = 24

step-by-step explanation:

a)

if joe plays 5 games, this will imply that x = 5. substitute x = 5 in the equation and solve for y;

y = -.50x + 12

y = -0.50(5) + 12

y = 9.50

b)

if joe leaves the arcade with $3.50, this will mean that y = 3.50. plug in y = 3.50 in the given equation and solve for x;

y = -.50x + 12

3.50 = -0.50x + 12

0.50x = 12 - 3.50

0.5x = 8.5

x = 17

c)

the function given is   y = -.50x + 12.

the y-intercept is the point where the graph of this function will intersect the y-axis. at this point, the value of x is usually 0. plug in x = 0 in the given equation and determine y;

y = -0.50(0) + 12

y = 12

the y-intercept is thus (0, 12). this is the initial amount of money that joe had prior to playing any game.

d)

the x-intercept refers to the point where the graph of a function intersects with the x-axis. at this point, the value of y is 0. substitute y = 0 in the given equation and solve for x;

0 =   -.50x + 12

0.5x = 12

x = 24

x-intercept ( 24, 0). this is the number of games that joe will play before he depletes the initial amount of money that he had.

e)

the slope of the given equation is -0.5. the equation has been given in slope-intercept form. thus the slope is simply the coefficient of x.

joe's balance will reduce by 0.5 for every game that he plays. this is the interpretation of the slope.

f)

domain is the set of x values for which the function is real and defined. the domain will thus be; [0,24]. this will restrict the balance to only assume positive values.

on the other hand, the range is the set of y values for which a function is real and defined. the range in our case will be [0,12]. joe can have between $0 and $12 balance.

g)

the standard form of the equation of a straight line is;

ax + by = c

where a, b, and c are integers.

our equation is given as;

y = -.50x + 12

re-writing the equation we have;

0.5x + y = 12

divide all through by 0.5;

x + 2y = 24

b

step-by-step explanation:

the equation of a line in point- slope form is

y - b = m(x - a)

where m is the slope and (a, b) a point on the line

calculate m using the slope formula

m = (y₂ - y₁ ) / (x₂ - x₁ )

with (x₁, y₁ ) = (0, 4) and (x₂, y₂ ) = (3, - 5) ← 2 points on the line

m = = = - 3

with m = - 3 and (a, b) = (3, - 5), then

y - (- 5) = - 3(x - 3), that is

y + 5 = - 3(x - 3) → b