Please give an explanation? 30 points
1 answer:
The composition of two rotations about the same point (Z) is the sum of the rotations. This can be proved using the matrix versions of the rotations.
This means
where
R=rotation operator
z=centre of rotation
t1,t2 are angles of rotation
In this particular case, the composition is commutative, i.e. order does not matter because addition of the angles is also commutative.
Thus substituting values, t1=170, t2=100
So the result of the composition is a rotation of 270 degrees about Z from the original position, which is also equal to a rotation of -90 degrees.
This means
where
R=rotation operator
z=centre of rotation
t1,t2 are angles of rotation
In this particular case, the composition is commutative, i.e. order does not matter because addition of the angles is also commutative.
Thus substituting values, t1=170, t2=100
So the result of the composition is a rotation of 270 degrees about Z from the original position, which is also equal to a rotation of -90 degrees.
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Answer:
A. Null and alternative hypothesis:
B. Yes. At a significance level of 0.05, there is enough evidence to support the claim that there is signficant difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.
P-value = 0.00002
C. As the difference is calculated as (population 1 − population 2), being population 1: groceries and population 2: dinning out, and knowing there is evidence that the true mean difference is positive, we can say that the groceries annual credit card charge is higher than dinning out annual credit card charge.
The point estimate is the sample mean difference d=$840.
The 95% confidence interval for the mean difference between the population means is (490, 1190).
Step-by-step explanation:
This is a hypothesis test for the population mean.
The claim is that there is signficant difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.
Then, the null and alternative hypothesis are:
The significance level is 0.05.
The sample has a size n=42.
The sample mean is M=840.
As the standard deviation of the population is not known, we estimate it with the sample standard deviation, that has a value of s=1123.
The estimated standard error of the mean is computed using the formula:
Then, we can calculate the t-statistic as:
The degrees of freedom for this sample size are:
This test is a two-tailed test, with 41 degrees of freedom and t=4.848, so the P-value for this test is calculated as (using a t-table):
As the P-value (0.00002) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
At a significance level of 0.05, there is enough evidence to support the claim that there is signficant difference between the population mean credit card charges for groceries and the population mean credit card charges for dining out.
We have to calculate a 95% confidence interval for the mean difference.
The t-value for a 95% confidence interval and 41 degrees of freedom is t=2.02.
The margin of error (MOE) can be calculated as:
Then, the lower and upper bounds of the confidence interval are:
The 95% confidence interval for the mean difference is (490, 1190).