Which sequence of transformations will change figure PQRS to figure P′Q′R′S′?Counterclockwise rotation about the origin by 90 de
grees followed by reflection about the x-axis
Counterclockwise rotation about the origin by 90 degrees followed by reflection about the y-axis
Counterclockwise rotation about the origin by 180 degrees followed by reflection about the x-axis
Counterclockwise rotation about the origin by 180 degrees followed by reflection about the y-axis
Counterclockwise rotation about the origin by 90 degrees followed by reflection about the y-axis
Counterclockwise rotation about the origin by 180 degrees followed by reflection about the x-axis
Counterclockwise rotation about the origin by 180 degrees followed by reflection about the y-axis
2 answers:
Answer: The correct option is
(A) Counterclockwise rotation about the origin by 90 degrees followed by reflection about the x-axis
Step-by-step explanation: We are given to select the sequence of transformations that will change figure PQRS to figure P'Q'R'S'.
From the graph, we note that
the coordinates of the vertices of quadrilateral PQRS are P(-3, -2), Q(-2, -3), R(-3, -4) and S(-4, -4).
And, the co-ordinates of the vertices of quadrilateral P'Q'R'S' are P'(2, 3), Q'(3, 2), R'(4, 3) and S'(4, 4).
We see that
if a point (x, y) is rotated counterclockwise about the origin by 90 degrees and then reflected about the x-axis, its co-ordinates changes as follows :
So with the same sequence of transformations, we see that
P(-3, -2) ⇒ P'(2, 3),
Q(-2, -3) ⇒ Q'(3, 2),
R(-3, -4) ⇒ R'(4, 3)
and
S(-4, -4) ⇒ S'(4, 4).
Thus, the required sequence of transformations is
Counterclockwise rotation about the origin by 90 degrees followed by reflection about the x-axis
Option (A) is CORRECT.
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Answer:
a) One-tailed t-test about a mean.
b) The null hypothesis would be that there is no significant evidence to conclude that the adjusted grip leads to more distance on the thrown ball.
While the alternative hypothesis is that there is significant evidence to conclude that the adjusted grip leads to more distance on the thrown ball.
Mathematically,
The null hypothesis is represented as
H₀: μ ≤ 40 yards
The alternative hypothesis is given as
Hₐ: μ > 40 yards
c) p-value = 0.00621
d) The graph is presented in the attached image. The p-value obtained is less than the significance level at which the test was performed, hence, we accept the alternative hypothesis and say that there is significant evidence to conclude that the adjusted grip leads to more distance on the thrown ball.
Step-by-step explanation:
a) This test checks if the adjusted grip leads to more distance than before on the ball throws.
So, it is a one tailed test (testing only in one direction), a t-test about a mean.
b) For hypothesis testing, the first thing to define is the null and alternative hypothesis.
The null hypothesis plays the devil's advocate and usually takes the form of the opposite of the theory to be tested. It usually contains the signs =, ≤ and ≥ depending on the directions of the test.
While, the alternative hypothesis usually confirms the the theory being tested by the experimental setup. It usually contains the signs ≠, < and > depending on the directions of the test.
Since this question aims to test if adjusting the grip leads to more distance on the ball thrown.
The null hypothesis would be that there is no significant evidence to conclude that the adjusted grip leads to more distance on the thrown ball. That is, The adjusted grip leads to a distance lesser than or equal to the previous distance on the thrown ball.
While the alternative hypothesis is that there is significant evidence to conclude that the adjusted grip leads to more distance on the thrown ball.
Mathematically,
The null hypothesis is represented as
H₀: μ ≤ 40 yards
The alternative hypothesis is given as
Hₐ: μ > 40 yards
c) To do this test, we will use the z-distribution because there is information on the population standard deviation.
So, we compute the test statistic
z = (x - μ₀)/σ
x = 45 yards
μ₀ = 40 yards
σ = standard deviation = 2 yards
z = (45 - 40)/2 = 2.50
The p-value for a one-tailed test for z-test statistic of 2.50 is 0.00621
d) The sketch of the graph is presented in the attached image.
The interpretation of p-values is that
When the (p-value > significance level), we fail to reject the null hypothesis and when the (p-value < significance level), we reject the null hypothesis and accept the alternative hypothesis.
So, for this question, significance level = 0.05
p-value = 0.00621
0.00621 < 0.05
Hence,
p-value < significance level
This means that we reject the null hypothesis & accept the alternative hypothesis and say that there is enough evidence to conclude that the adjusted grip leads to more distance on the thrown ball.
Hope this Helps!!!
