Mathematics, 27.01.2020 21:31, Cassandra100

One hundred students were given an algebra test. a random sample of ten students was taken out of class of 800 enrolled students. the time it took each student to complete the test is recorded below. 22.2, 23.7, 16.8, 18.3, 19.7, 16.9, 17.2, 18.5, 21.0, and 19.7

a. find the mean, variance and standard deviation for this sample of ten students

b. construct a 95% confidence interval for the population mean time to complete this algebra test.

c. test if the population mean time to complete the test is 22.5 minutes.

Answers: 1

Show answers

Answers

answered: alisa1600000

a) $\bar X= \frac{\sum_{i=1}^n X_i}{10} =19.4$

$s^2 = \frac{\sum_{i=1}^n (x_i -\bar x)^2}{n-1}=5.438$

$s= \sqrt{5.438}=2.332$

b) $19.4-2.262\frac{2.332}{\sqrt{10}}=17.732$

$19.4+2.262\frac{2.332}{\sqrt{10}}=21.068$

So on this case the 95% confidence interval would be given by (17.732;21.068)

c) $t=\frac{19.4-22.5}{\frac{2.332}{\sqrt{10}}}=-4.203$

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

$p_v =2*P(t_{9}$

If we compare the p value and the significance level assumed $\alpha=0.05$ we see that p_v so we can conclude that we have enough evidence to reject the null hypothesis, so we can conclude that the true mean is significantly different from 22.5 minutes.

Step-by-step explanation:

Part a

We have the following data given: 22.2, 23.7, 16.8, 18.3, 19.7, 16.9, 17.2, 18.5, 21.0, and 19.7

We can calculate the sample mean with this formula:

$\bar X= \frac{\sum_{i=1}^n X_i}{10} =19.4$

And the sample variance with this formula:

$s^2 = \frac{\sum_{i=1}^n (x_i -\bar x)^2}{n-1}=5.438$

And the sample deviation would be just the square root of the sample variance

$s= \sqrt{5.438}=2.332$

Part b

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

$\bar X =19.4$ represent the sample mean

$\mu$ population mean (variable of interest)

s=2.332 represent the sample standard deviation

n=10 represent the sample size

The confidence interval for the mean is given by the following formula:

$\bar X \pm t_{\alpha/2}\frac{s}{\sqrt{n}}$ (1)

In order to calculate the critical value $t_{\alpha/2}$ we need to find first the degrees of freedom, given by:

df=n-1=10-1=9

Since the Confidence is 0.95 or 95%, the value of $\alpha=0.05$ and $\alpha/2 =0.025$ , and we can use excel, a calculator or a table to find the critical value. The excel command would be: "=-T.INV(0.025,9)".And we see that $t_{\alpha/2}=2.262$

Now we have everything in order to replace into formula (1):

$19.4-2.262\frac{2.332}{\sqrt{10}}=17.732$

$19.4+2.262\frac{2.332}{\sqrt{10}}=21.068$

So on this case the 95% confidence interval would be given by (17.732;21.068)

Part c

Null hypothesis: $\mu =22.5$

Alternative hypothesis: $\mu \neq 22.5$

Since we don't know the population deviation, is better apply a t test to compare the actual mean to the reference value, and the statistic is given by:

$t=\frac{\bar X-\mu_o}{\frac{s}{\sqrt{n}}}$ (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".

Calculate the statistic

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

$t=\frac{19.4-22.5}{\frac{2.332}{\sqrt{10}}}=-4.203$

P-value

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

$p_v =2*P(t_{9}$

Conclusion

answered: Guest

answer: on kudzu, being female and being green are independent events.

step-by-step explanation:

answered: Guest

answer: group b.

step-by-step explanation:

Other questions on the subject: Mathematics

Mathematics, 21.06.2019 13:00, stormhorn491

Which of the following is the equation of a line that passes through the point (1,4) and is parallel to the x-axis a. y=1 b. y=4 c. x=1 d. x=4

Answers: 1

continue

Mathematics, 21.06.2019 18:30, lionscoachjose

Tod does not have any cookies. david gives jeff 8 cookies. then he splits half of the cookies he has left with tod. david let’s c represent the number of cookies that he starts with. he finds the number of cookies that tod has is 1/2 the difference of c and 8. write an expression to represent the number of cookies that tod has.

Answers: 2

continue

Mathematics, 21.06.2019 22:40, Savannahh8503

Aclassmate thinks that solving a system by graphing gives an exact answer when the lines appear to cross at a grid point, but only an approximate answer when they don't. explain why this isn't true.

Answers: 3

continue

Mathematics, 22.06.2019 01:30, haleymoodie6034

Mrs. julien’s and mrs. castillejo’s classes are selling cookie dough for a school fundraiser. customers can buy packages of macadamia nut chip cookie dough and packages of triple chocolate cookie dough. mrs. julien’s class sold 25 packages of macadamia nut chip cookie dough and 30 packages of triple chocolate cookie dough for a total of $221.25. mrs. castillejo’s class sold 5 packages of macadamia nut chip cookie dough and 45 packages of triple chocolate cookie dough for a total of $191.25. (a) write the system of equations that model the problem. be sure to explain which equation represents which situation. (b) find the cost of each type of cookie. show your work. (c) explain which method you used to solve the system and why you chose that method.

Answers: 2

continue

Do you know the correct answer?

One hundred students were given an algebra test. a random sample of ten students was taken out of cl...