please show work thank you!

in Statistics Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

We need the Z score for each of the grade conditions.

Z=(X-µ)/σ gives the relevant Z score. Let’s apply this formula for each grade where X=x. Ignore the inequality for the time being and just use equality. We can come back to the inequality later.

A: x=µ+σ, Z=(µ+σ-µ)/σ=1, P(Z=1)=0.8413.

B: x=µ+σ and x=µ, Z₁=1, P₁(Z₁=1)=0.8413 and Z₂=0, P₂(Z₂=0)=0.5.

C: x=µ-σ and x=µ, Z₁=(µ-σ-µ)/σ=-1, P₁(Z₁=-1)=1-P(Z=1)=1-0.8413=0.1587, P₂(Z₂=0)=0.5.

D: x=µ-2σ and x=µ-σ, Z₁=(µ-2σ-µ)/σ=-2, P₁(Z₁=-2)=1-P(Z=2)=1-0.9772=0.0228, P₂(Z₂=-1)=0.1587.

F: x=µ-2σ, P(Z=-2)=0.0228.

Now, let’s analyse what we just did.

Here’s the nice bit—a picture.

The red bell-shaped curve is the normal distribution. The vertical lines represent the Z scores for the grades, or, rather, the cutoff or critical values that separate the grades, because a grade is a band (determined by the inequalities in the question). The area under the red curve to the right of the green line is the grade A band; the area between the amber and green lines is the grade B band; the area between the red and amber lines is the grade C band; and the the area between the blue and red lines is the grade D band. The area to the left of the blue band is all grade F.

It should now be getting clear what to do next. The total area under the curve is 100% of all the results, so we can see that each grade band is a fraction of this area, that is, some percentage. Normal distribution tables give the area under the curve to the the left (that is, less than) a specific Z score.

For grade A we need the area to the right, so we can get this in two ways. Look up Z=1 to get the area to the left, then subtract this from 1 (or 100%) to get the area to the right. The other way makes use of symmetry. The area to the left of Z=-1 has the same size as the area to the right of Z=1. So the band for grade A is 1-0.8413=0.1587 or 15.87%.

Note that the amber line splits the distribution in half exactly, and corresponds to 0.5 or 50% of the results. We know the area to the left of the green line is 0.8413 and the area to the left of the amber line is 0.5. So the area in between is 0.8413-0.5=0.3413 or 34.13%. Grade B is therefore about 34%.

Grade C band is the mirror of grade B band, so must also be 34%.

We know that the area to the left of the red line is the same size as the area to the right of the green line, 0.1587. This is the upper limit for the grade D band. The lower limit is when Z=-2, corresponding to 0.0228. The difference between these is 0.1587-0.0228=0.1359 or 13.59%, and this is the percentage for grade D.

Grade F is to the left of the blue line at Z=-2, and consists of 0.0228 or 2.28% of the results.

I hope this detailed explanation helps you to understand how to solve this type of problem.

 

by Top Rated User (1.1m points)

Related questions

0 answers
1 answer
1 answer
asked Jun 5, 2013 in Statistics Answers by anonymous | 454 views
1 answer
1 answer
2 answers
1 answer
1 answer
asked Aug 17, 2020 in Statistics Answers by anonymous | 821 views
Welcome to MathHomeworkAnswers.org, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!

Most popular tags

algebra problems solving equations word problems calculating percentages math problem geometry problems calculus problems math fraction problems trigonometry problems rounding numbers simplifying expressions solve for x order of operations probability algebra pre algebra problems word problem evaluate the expression slope intercept form statistics problems factoring polynomials solving inequalities 6th grade math how to find y intercept equation of a line sequences and series algebra 2 problems logarithmic equations solving systems of equations by substitution dividing fractions greatest common factor square roots geometric shapes graphing linear equations long division solving systems of equations least to greatest dividing decimals substitution method proving trigonometric identities least common multiple factoring polynomials ratio and proportion trig identity precalculus problems standard form of an equation solving equations with fractions http: mathhomeworkanswers.org ask# function of x calculus slope of a line through 2 points algebraic expressions solving equations with variables on both sides college algebra domain of a function solving systems of equations by elimination differential equation algebra word problems distributive property solving quadratic equations perimeter of a rectangle trinomial factoring factors of a number fraction word problems slope of a line limit of a function greater than or less than geometry division fractions how to find x intercept differentiation exponents 8th grade math simplifying fractions geometry 10th grade equivalent fractions inverse function area of a triangle elimination method story problems standard deviation integral ratios simplify systems of equations containing three variables width of a rectangle percentages area of a circle circumference of a circle place value solving triangles parallel lines mathematical proofs solving linear equations 5th grade math mixed numbers to improper fractions scientific notation problems quadratic functions number of sides of a polygon length of a rectangle statistics zeros of a function prime factorization percents algebra 1 evaluating functions derivative of a function equation area of a rectangle lowest common denominator solving systems of equations by graphing integers algebra 2 diameter of a circle dividing polynomials vertex of a parabola calculus problem perpendicular lines combining like terms complex numbers geometry word problems converting fractions to decimals finding the nth term range of a function 4th grade math greatest to least ordered pairs functions radius of a circle least common denominator slope unit conversion solve for y calculators solving radical equations calculate distance between two points area word problems equation of a tangent line multiplying fractions chemistry binomial expansion place values absolute value round to the nearest tenth common denominator sets set builder notation please help me to answer this step by step significant figures simplifying radicals arithmetic sequences median age problem trigonometry graphing derivatives number patterns adding fractions radicals midpoint of a line roots of polynomials product of two consecutive numbers limits decimals compound interest please help pre-algebra problems divisibility rules graphing functions subtracting fractions angles numbers discrete mathematics volume of a cylinder simultaneous equations integration probability of an event comparing decimals factor by grouping vectors percentage expanded forms rational irrational numbers improper fractions to mixed numbers algebra1 matrices logarithms how to complete the square mean statistics problem analytic geometry geometry problem rounding decimals 5th grade math problems solving equations with variables solving quadratic equations by completing the square simplifying trigonometric equation using identities
87,441 questions
99,039 answers
2,422 comments
16,939 users