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2 years ago

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A curious student in a large economics course is interested in calculating the percentage of his classmates who scored lower tha

n he did on the GMAT; he scored 490. He knows that GMAT scores are normally distributed and that the average score is approximately 540. He also knows that 95% of his classmates scored between 400 and 680. Based on this information, calculate the percentage of his classmates who scored lower than he did?

1 answer:

LuckyWell [14K]2 years ago

6 0

Answer: 23.89%

Step-by-step explanation:

The empirical rule says that the 95% of the data falls in between two standard deviations of the mean.

Given : GMAT scores are normally distributed and the the average score is approximately $\mu=540$ .

Also, 95% of his classmates scored between 400 and 680.

Then, by empirical rule , 95% of data falls in between $mu\pm 2\sigma$

i.e. $540- 2\sigma=400$ (1)

$540+2\sigma=680$ (2)

Subtracting (1) from (2), we get

$4\sigma=680-400=280\\\\\Rightarrow\ \sigma=\dfrac{280}{4}=70$

Let x be the random variable to represent the scores of every student.

Statistic z-score : $z=\dfrac{x-\mu}{\sigma}$

For x= 490, we have

$z=\dfrac{490-540}{70}\approx-0.71$

The p-value = $P(z$

Hence, 23.89% of his classmates who scored lower than he did .

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Then multiply:
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Now to check:
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2 years ago

Test the given claim. Identify the null hypothesis, alternative hypothesis, test statistic, P-value, and then state the concl

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Answer:

a) Failed to reject the null hypothesis (P-value=0.09).

b) The 95% CI for the difference in proportions is:

$-0.0599\leq\pi_1-\pi_2\leq0.0124$

Step-by-step explanation:

a) We have to perform a hypothesis test for the difference of proportions.

The null and alternative hypothesis are:

$H_0: \pi_1\geq\pi_2\\\\H_1: \pi_1$

The significance level is 0.05.

The proportion of the passenger cars owners is:

$p_1=\frac{239}{2142} =0.1116$

The proportion of commercial truck owners is:

$p_2=\frac{54}{399}=0.1353$

The weigthed average p is

$p=\frac{n_1p_1+n_2p_2}{n_1+n_2}=\frac{239+54}{2142+399}=0.1153$

The estimated standard deviation is

$s=\sqrt{\frac{p(1-p)}{n_1}+\frac{p(1-p)}{n_2}} =\sqrt{\frac{0.1153(1-0.1153)}{2142}+\frac{0.1153(1-0.1153)}{399}} =0.0174$

We can calculate the z-value as:

$z=\frac{\Delta p}{s}=\frac{0.1116-0.1353}{0.0174}=-1.362$

The P-value for z=-1.362 is P=0.0866.

The P-value (0.09) is greater than the significance level (0.05), so it failed to reject the null hypothesis. There is no enough evidence to prove that commercial trucks owners violate laws requiring front license plates at a higher rate than owners of passenger cars.

b) We can construct a 95% CI, according to the significance level of 0.05.

The z-value for this CI is 1.96.

We have to recalculate the standard deviation:

$\sigma=\sqrt{\frac{p_1(1-p_1)}{n_1} +\frac{p_2(1-p_2)}{n_2}} =\sqrt{\frac{0.1116(1-0.1116)}{2142} +\frac{0.1353(1-0.1353)}{399}} =0.0184$

The lower limit is then:

$LL=(p_1-p_2)-z*\sigma=(0.1116-0.1353)-1.96*0.0184=-0.0238-0.0361\\\\LL=-0.0599$

The upper limit is:

$UL=(p_1-p_2)+z*\sigma=(0.1116-0.1353)+1.96*0.0184=-0.0238+0.0361\\\\UL=0.0124$

The 95% CI for the difference in proportions is:

$-0.0599\leq\pi_1-\pi_2\leq0.0124$

In this case, we can conclude that the difference between the proportions, with 95% confidence, can still be equal or greater than zero, meaning that it is possible passenger car owners violate laws more than truck owners.

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2 years ago

The volleyball team at West View High School is comparing T-shirt companies where they can purchase their practice shirts. The t

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10.5x = 7.5x + 30
10.5x - 7.5x = 30
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x = 10

10.5(10) = 105
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they need to purchase 10 t-shirts from each company for a total of $105 for each company

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2 years ago

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Mrs. Ishimitsu is installing a rubber bumper around the edge of her coffee table. The dimensions of the rectangular table are (2

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Answer:

$2x^2+8x-30$ represents the total perimeter of the table, and if x = 3 the lentgh of entire rubber bumper is 12 feet.

Step-by-step explanation:

Perimeter is defined as the sum of sides.

For a rectangle having dimension length 'L' and breadth 'B' Since, rectangle has two sides equal.

Perimeter of rectangle = 2 ( length + breadth )

Here, given rectangle has dimension

Length = $2x^2-16$ and breadth = $-x^2+4x+1$

Thus, perimeter of rectangle = $2(2x^2-16+(-x^2+4x+1))$

$=2(2x^2-16-x^2+4x+1)$

$=2(x^2+4x-15)$

$=2x^2+8x-30$

Thus, perimeter of rectangle is $2x^2+8x-30$ ........(1)

Perimeter of rectangle , when x = 3,

Put x= 3 in (1) ,

$2x^2+8x-30\Rightarrow 2(3)^2+8(3)-30 \Rightarrow 18+24-30=12$

Thus, $2x^2+8x-30$ represents the total perimeter of the table, and if x = 3 the lentgh of entire rubber bumper is 12 feet.

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2 years ago

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Emily is a songwriter who collects royalties on her songs whenever they are played in a commercial or a movie. Emily will earn $

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Answer:

(6,2)

Step-by-step explanation:

Variable Definitions:

x= the number of commercials

y= the number of movies

Each commercial earns Emily $50, so x commercials would earn her 50x dollars in royalties. Each movie earns Emily $150, so y movies would earn her 150y dollars in royalties. Therefore, the total royalties 50x+150y equals $600:

50x+150y=600

Since Emily's songs were played on 3 times as many commercials as movies, if we multiply 3 by the number of movies, we will get the number of commercials, meaning x equals 3y.

x=3y

Write System of Equations:

50x+150y=600

x=3y

Solve for y in each equation:

1) 50x+150y=600

150y=−50x+600

y=-1/3x+4

2) x=3y

y=1/3x

The x variable represents the number of commercials and the yy variable represents the number of movies. Since the lines intersect at the point (6,2) we can say:

Emily's songs were played on 6 commercials and 2 movies.

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2 years ago