Item 4

At a large Midwestern university, a simple random sample of 100 entering freshmen in 1993 found that 20 of the sampled freshmen

guajiro [1.7K]

Answer:

The 90% confidence interval for the difference of proportions is (0.01775,0.18225).

Step-by-step explanation:

Before building the confidence interval, we need to understand the central limit theorem and subtraction of normal variables.

Central Limit Theorem

The Central Limit Theorem establishes that, for a normally distributed random variable X, with mean $\mu$ and standard deviation $\sigma$ , the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean $\mu$ and standard deviation $s = \frac{\sigma}{\sqrt{n}}$ .

For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.

For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean $\mu = p$ and standard deviation $s = \sqrt{\frac{p(1-p)}{n}}$

Subtraction between normal variables:

When two normal variables are subtracted, the mean is the difference of the means, while the standard deviation is the square root of the sum of the variances.

p1 -> 1993

20 out of 100, so:

$p_1 = \frac{20}{100} = 0.2$

$s_1 = \sqrt{\frac{0.2*0.8}{100}} = 0.04$

p2 -> 1997

10 out of 100, so:

$p_2 = \frac{10}{100} = 0.1$

$s_2 = \sqrt{\frac{0.1*0.9}{100}} = 0.03$

Distribution of p1 – p2:

$p = p_1 - p_2 = 0.2 - 0.1 = 0.1$

$s = \sqrt{s_1^2+s_2^2} = \sqrt{0.04^2 + 0.03^2} = 0.05$

Confidence interval:

$p \pm zs$

In which

z is the z-score that has a p-value of $1 - \frac{\alpha}{2}$ .

90% confidence level

So $\alpha = 0.1$ , z is the value of Z that has a p-value of $1 - \frac{0.1}{2} = 0.95$ , so $Z = 1.645$ .

The lower bound of the interval is:

$p - zs = 0.1 - 1.645*0.05 = 0.01775
$

The upper bound of the interval is:

$p + zs = 0.1 + 1.645*0.05 = 0.18225
$

The 90% confidence interval for the difference of proportions is (0.01775,0.18225).

6 0

3 years ago

Least common multiple of 70, 60, and 50

goldenfox [79]

70 = 7 x 2 x 5
60 = 2² x 3 x 5
50 = 2 x 5²

LCM = 2 x 3 x 5 x 7 = 210

Answer: LCM = 210

3 0

3 years ago

Subtract -5x^2+7x+1 from 8x^2+x-10<br>I will give brainleist

marta [7]

Answer:

3x² + 8x - 9

Step-by-step explanation:

8x² - 5x² +x+7x-10+1

=3x² + 8x - 9

8 0

3 years ago

Drag the tiles to the boxes to form correct pairs. Match each function to the equation of the tangent line to its graph at x = 2

Ede4ka [16]

Step-by-step explanation:

the picture includes the answer and work

I plugged in 2 for x

the equations that got the same answer, I put together

3 0

3 years ago

Helpppp meeeee outttttt pleaseeeee ASAPPPP!!!!

Ilya [14]

Answer:

$\boxed{\sf sin\ C =\frac{40}{41}}$

Step-by-step explanation:

We need to find out the value of sinC using the given triangle . Here we can see that the sides of the triangle are 40 , 41 and 9 .

We know that the ratio of sine is perpendicular to hypontenuse .

$\sf\longrightarrow sin\theta =\dfrac{ perpendicular}{hypontenuse}$

Here we can see that the side opposite to angle C is 40 , therefore the perpendicular of the triangle is 40. And the side opposite to 90° angle is 41 . So it's the hypontenuse . On using the ratio of sine ,

$\sf\longrightarrow sinC =\dfrac{ p}{h}\\\\\sf\longrightarrow sin\ C =\dfrac{AB}{AC}$

Substitute the respective values ,

$\sf\longrightarrow \boxed{\blue{\sf sin\ C =\dfrac{40}{41}}}$

<u>Hence the required answer is 40/41.</u>

6 0

3 years ago