A city is interested in building a parking garage in a certain area. Three hundred randomly selected residents from this area were asked, "Do you support the city's decision to build a parking garage in your area?" Of the 300 residents interviewed, 203 said no, 7 said yes, and 90 had no opinion. A large-sample z-confidence interval, p ± z * square root of p hat times 1 minus p hat over n, was constructed from these data to estimate the proportion of the residents who support building a parking garage in a certain area. Which of the following statements is correct for this confidence interval?
This confidence interval is valid because a sample size of more than 30 was used.
This confidence interval is not valid because the quantity np is too small.
This confidence interval is valid because each area resident was asked the same question.
The confidence interval is valid because no conditions are required for constructing a large-sample confidence interval for a proportion.
This confidence interval is not valid because "no opinion" was included as a response category for the question.

Answer:

This confidence interval is not valid because the quantity np is too small.

Step-by-step explanation:

300(7/300)= 7<10

The measure of the length and width of the rectangle is given as 135 feet and 92 feet respectively.

A rectangle is a quadrilateral with four right angles. It can also be defined as: an equiangular quadrilateral, since equiangular means that all of its angles are equal; or a parallelogram containing a right angle. A rectangle with four sides of equal length is a square.

Here, we have,

to determine the dimensions

From the information given, we have the following proofs;

Width, w = x+15

Length, l = 2(x+15) - 49

Length, l = 2x + 30 - 49

Length = 2x - 19

The formula for perimeter of a rectangle is given as;

Perimeter = 2( length + width)

Substitute the expressions into the formula

Perimeter =  2 ( x+ 15 + 2x - 19 )

Perimeter = 2 (3x - 4)

Perimeter = 6x - 8

We have that the area is  162 more than 27 times the perimeter, which is Area = 27 (perimeter )+ 163

Area = 27(6x-8) + 162

Expand the bracket

Area= 162x - 216 + 162

Area = 162x - 54

But we know that

Area = length × width

Substitute the expressions

Area = (x+15)(2x-19)

Area =  2x² - 19x +30x - 285

Area = 2x² + 11x - 285

Equate the two formulas for area

162x - 54 = 2x² +11x - 285

Collect like terms

2x² + 11x - 285 - 162x + 54 = 0

2x²  - 151x - 231 = 0

Solve the quadratic equation

(2x + 3)(x-77) = 0

Let's solve for x

x - 77 = 0

x = 77

The expression for the width;

Width = x+15

Width = 77 + 15

Width = 92 feet

The expression for the length

Length = 2(x+15) - 49

Length = 2 ( 77 + 15) - 49

Length = 154 + 30 - 49

Length = 184 - 49

Length = 135 feet

Thus, the measure of the length and width of the rectangle is given as 135 feet and 92 feet respectively.

Learn more about a rectangle here:

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