Step-by-step explanation:
Zero has one square root which is 0. Negative numbers don't have real square roots since a square is either positive or 0. The square roots of numbers that are not a perfect square are members of the irrational numbers. This means that they can't be written as the quotient of two integers.
An economist is studying the job market in Denver area neighborhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs).x 15 32 51 28 50 25y 3 3 7 5 9 3Complete parts (a) through (e), given Σx = 201, Σy = 30, Σx2 = 7759, Σy2 = 182, Σxy = 1163, and r ≈ 0.872.a. Draw a scatter diagram displaying the data.b. Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r.c. Find x, and y. Then find the equation of the least-squares line = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.)d. Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.
Answer:
The sample correlation coefficient is, r = 0.8722.
The equation of the least-squares line is:
[tex]y= -0.161+0.154x[/tex]
Step-by-step explanation:
(a)
The scatter diagram displaying the data for X : total number of jobs in a given neighborhood and Y : number of entry-level jobs in the same neighborhood is shown below.
(b)
The table attached below verifies the values of [tex]\sum X,\ \sum Y,\ \sum X^{2},\ \sum Y^{2}\ \text{and}\ \sum XY[/tex].
The sample correlation coefficient is:
[tex]\begin{aligned}r~&=~\frac{n\cdot\sum{XY} - \sum{X}\cdot\sum{Y}} {\sqrt{\left[n \sum{X^2}-\left(\sum{X}\right)^2\right] \cdot \left[n \sum{Y^2}-\left(\sum{Y}\right)^2\right]}} \\r~&=~\frac{ 6 \cdot 1163 - 201 \cdot 30 } {\sqrt{\left[ 6 \cdot 7759 - 201^2 \right] \cdot \left[ 6 \cdot 182 - 30^2 \right] }} \approx 0.8722\end{aligned}[/tex]
Thus, the sample correlation coefficient is, r = 0.8722.
(c)
The slope and intercept are:
[tex]\begin{aligned} a &= \frac{\sum{Y} \cdot \sum{X^2} - \sum{X} \cdot \sum{XY} }{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 30 \cdot 7759 - 201 \cdot 1163}{ 6 \cdot 7759 - 201^2} \approx -0.161 \\ \\b &= \frac{ n \cdot \sum{XY} - \sum{X} \cdot \sum{Y}}{n \cdot \sum{X^2} - \left(\sum{X}\right)^2} = \frac{ 6 \cdot 1163 - 201 \cdot 30 }{ 6 \cdot 7759 - \left( 201 \right)^2} \approx 0.154\end{aligned}[/tex]
The equation of the least-squares line is:
[tex]y= -0.161+0.154x[/tex]
(d)
The least-squares line is graphed in the diagram below.
"National survey released in 2003 showed that among U.S. adults ages 70 and older, 21.1% had been told by a doctor that they had some form of cancer. Use this result as valid for the population of U.S. adults, 70 yrs. old and older. What is the probability that among 40 adults (ages 70 and older) chosen at random more than 22 percent will have been told by their doctor that they have cancer
Answer:
44.43% probability that among 40 adults (ages 70 and older) chosen at random more than 22 percent will have been told by their doctor that they have cancer
Step-by-step explanation:
To solve this question, we need to understand the normal probability distribution and the central limit theorem.
Normal probability distribution
When the distribution is normal, we use the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.
Central Limit Theorem
The Central Limit Theorem estabilishes that, for a normally distributed random variable X, with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the sampling distribution of the sample means with size n can be approximated to a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]s = \frac{\sigma}{\sqrt{n}}[/tex].
For a skewed variable, the Central Limit Theorem can also be applied, as long as n is at least 30.
For a proportion p in a sample of size n, the sampling distribution of the sample proportion will be approximately normal with mean [tex]\mu = p[/tex] and standard deviation [tex]s = \sqrt{\frac{p(1-p)}{n}}[/tex]
In this question, we have that:
[tex]p = 0.211, n = 40[/tex]
So
[tex]\mu = 0.211, s = \sqrt{\frac{0.211*0.789}{40}} = 0.0645[/tex]
What is the probability that among 40 adults (ages 70 and older) chosen at random more than 22 percent will have been told by their doctor that they have cancer
This is 1 subtracted by the pvalue of Z when X = 22. So
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
By the Central Limit Theorem
[tex]Z = \frac{X - \mu}{s}[/tex]
[tex]Z = \frac{0.22 - 0.211}{0.0645}[/tex]
[tex]Z = 0.14[/tex]
[tex]Z = 0.14[/tex] has a pvalue of 0.5557
1 - 0.5557 = 0.4443
44.43% probability that among 40 adults (ages 70 and older) chosen at random more than 22 percent will have been told by their doctor that they have cancer
A random sample of 1000 people was taken. Four hundred fifty of the people in the sample favored Candidate AT. The 95% confidence interval for the true proportion of people who favor Candidate A is a. .424 to .476. b. .419 to .481. c. .40 to .50. d. .45 to .55.
Answer:
[tex]0.45 - 1.96 \sqrt{\frac{0.45(1-0.45)}{1000}}=0.419[/tex]
[tex]0.45 + 1.96 \sqrt{\frac{0.45(1-0.45)}{1000}}=0.481[/tex]
And the 95% confidence interval would be given (0.419;0.481). And the best option would be:
b. .419 to .481
Step-by-step explanation:
We know the following info:
[tex]n = 1000[/tex] sample size selected
[tex]X= 450[/tex] represent the number of people who favored Candidate AT
The sample proportion would be:
[tex]\hat p=\frac{450}{1000}=0.45[/tex]
The confidence interval would be given by this formula
[tex]\hat p \pm z_{\alpha/2} \sqrt{\frac{\hat p(1-\hat p)}{n}}[/tex]
For the 95% confidence interval the value of [tex]\alpha=1-0.95=0.05[/tex] and [tex]\alpha/2=0.025[/tex], with that value we can find the quantile required for the interval in the normal standard distribution.
[tex]z_{\alpha/2}=1.96[/tex]
And replacing into the confidence interval formula we got:
[tex]0.45 - 1.96 \sqrt{\frac{0.45(1-0.45)}{1000}}=0.419[/tex]
[tex]0.45 + 1.96 \sqrt{\frac{0.45(1-0.45)}{1000}}=0.481[/tex]
And the 95% confidence interval would be given (0.419;0.481). And the best option would be:
b. .419 to .481
At the grocery store, Cory has narrowed down his selections to 6 vegetables, 6 fruits, 7 cheeses, and 5 whole grain breads. He wants to use the Express Lane, so he can only buy 15 items. In how many ways can he choose which 15 items to buy if he wants all 6 fruits?
Answer:
48620
Step-by-step explanation:
There are 6 fruits and 18 non-fruits. Cory wants to buy all 6 fruits, and 9 of the 18 non-fruits.
The number of ways he can choose 6 fruits from 6 is ₆C₆ = 1.
The number of ways he can choose 9 non-fruits from 18 is ₁₈C₆ = 48620.
The total number of combinations is 1 × 48620 = 48620.
Divide up the number 480 in a ratio of 3:5.
Answer:
180:300
Step-by-step explanation:
You first divide 480 by 8 because 3+5= 8 and then you multiply that answer (60) by 3 to get 180 and then you multiply it by 5 to get 300. So you get the ratio of 180:300.
Simply -5+2(x-3)+7x :)
Answer:
9x-11
Step-by-step explanation:
-5+2(x-3)+7x
Distribute
-5 +2x -6 +7x
-11 +9x
9x-11
Answer:
[tex]= 9x - 11 \\ [/tex]
Step-by-step explanation:
[tex] - 5 + 2(x - 3) + 7x \\ - 5 + 2x - 6 + 7x \\ - 5 - 6 + 2x + 7x \\ - 11 + 9x \\ = 9x - 11[/tex]
Please answer this correctly
Answer:
1/8 of the buckets
Step-by-step explanation:
There's one X for 1 1/2 cups which is greater than 1 1/4 cups but less than 1 3/4 cups. There are 8 pieces of data in total so our answer is 1/8 of the buckets.
eBookThe manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is or less. A member of the hotel's accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of future weekend guest bills to test the manager's claim.a. Which form of the hypotheses should be used to test the manager's claim
Answer:
Step-by-step explanation:
The question is incomplete. The complete question is
The manager of the Danvers-Hilton Resort Hotel stated that the mean guest bill for a weekend is $600 or less. A member of the hotel’s accounting staff noticed that the total charges for guest bills have been increasing in recent months. The accountant will use a sample of
weekend guest bills to test the manager’s claim.
a. Which form of the hypotheses should be used to test the manager’s claim? Explain.
1) H0:μ ≥ 600
Ha:μ < 600
2) H0:μ ≤ 600
Ha:μ > 600
3) H0:μ=600
Ha:μ≠600
b. What conclusion is appropriate when H0 cannot be rejected?
c. What conclusion is appropriate when H0 can be rejected?
Solution:
a) the hypotheses should be used to test the manager’s claim is
H0:μ ≤ 600
Ha:μ > 600
This is because the already known or assumed mean guest bill for a weekend is 600 or less. This forms the null hypothesis. The alternative is the opposite of the null hypothesis. Since the alternative states that it is increasing, the sign,> would be used.
b) If H0 cannot be rejected, it means that there is no sufficient evidence to reject H0 at the given level of significance.
c) if H0 can be rejected, it means that there is sufficient evidence to reject H0 at the given level of significance.
EXREAMLY URGENT!! WILL FOREVER THANK YOU!!!! PLS JUST TAKE A LOOK!!!!! 20. What is the area of triangle XWZ?
Answer:
72√3
Step-by-step explanation:
30 60 90 triangles are what you start out with.
Step 1: 30-60-90
x = 12
WZ = 12√3
Step 2: Area formula
A = 1/2(12)(12√3)
*Since the 2 30-60-90 triangles are congruent, both segments of the base are 12
Plug it into the calc and you should get A = 72√3 as your final answer!
A pastry chef is making a batch of mini petit fours, which are little cakes, in the shape of cubes. To keep the nutritional value of each petit four consistent, the bakery manager wants each one to have a volume of 45cm3. What should the side length be, to the nearest hundredth, for each petit four? (Note: For volume of a cube, V=s3 where s=side length.)
Answer:
3.56 cm
Step-by-step explanation:
Cube is a 3D closed structure in which each adjacent side is perpendicular to each other and every side is equal to each other.
Let the side of cube be [tex]a[/tex] cm.
Please refer to attached image of cube for a clear look and feel of a cube with each side = a units.
Then, volume of cube is given by the formula:
[tex]V = a^3[/tex]
Here, we are given that:
[tex]V = 45\ cm^3[/tex]
[tex]\Rightarrow a^3 = 45\ cm^3\\\Rightarrow a =\sqrt[3] {45}\\\Rightarrow a ={45}^\frac{1}{3}\\\Rightarrow a = 3.56\ cm[/tex]
So, the answer is, Side of each petit four is, [tex]a = 3.56\ cm[/tex].
Make the appropriate conclusion. Choose the correct answer below. A. RejectReject H0. There is insufficientinsufficient evidence at the alphaαequals=0.100.10 level of significance to conclude that the true mean heart rate during laughter exceeds 7171 beats per minute. B. Do not rejectDo not reject H0. There is insufficientinsufficient evidence at the alphaαequals=0.100.10 level of significance to conclude that the true mean heart rate during laughter exceeds 7171 beats per minute. C. RejectReject H0. There is sufficientsufficient evidence at the alphaαequals=0.100.10 level of significance to conclude that the true mean heart rate during laughter exceeds 7171 beats per minute. D. Do not rejectDo not reject H0. There is sufficientsufficient evidence at the alphaαequals=0.100.10 level of significance to conclude that the true mean heart rate during laughter exceeds 7171 beats per minute.
Answer:
a) Option D is correct.
H0: μ = 71
Ha: μ > 71
b) Option F is correct
z > 1.28
c) z = 2.85
d) Option C is correct.
Reject H0.There is sufficient evidence at the α = 0.10 level of significance to conclude that the true mean heart rate during laughter exceeds 71 beats per minute.
Step-by-step explanation:
a) 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.
This question aims to test the the true mean heart rate during laughter exceeds 71 beats per minute.
Hence, the null hypothesis is that there isn't sufficient evidence to say that the true mean heart rate during laughter exceeds 71 beats per minute. That is, the true mean doesn't exceed 71 beats per minute.
And the alternative hypothesis is that there is sufficient evidence to say that the true mean heart rate during laughter exceeds 71 beats per minute.
Mathematically,
The null hypothesis is represented as
H₀: μ = 71
The alternative hypothesis is represented as
Hₐ: μ > 71
b) Using z-distribution, the rejection area is obtained from the confidence level at which the test is going to be performed. Since the hypothesis test tests only in one direction,
Significance level = (100% - confidence level)/2
0.10 = 10% = (100% - confidence level)/2
20% = 100% - (confidence level)
Confidence level = 100% - 20% = 80%
Critical value for 80% confidence level = 1.28
And since we are testing if the true mean heart rate during laughter exceeds 71 beats per minute, the rejection area would be
z > 1.28
c) The test statistic is given as
z = (x - μ)/σₓ
x = sample mean = 73.4
μ = 71
σₓ = standard error = (σ/√n)
σ = 8
n = Sample size = 90
σₓ = (8/√90) = 0.8433
z = (73.4 - 71) ÷ 0.8433
z = 2.846 = 2.85
d) Since the z-test statistic obtained, 2.85, is firmly in the rejection area, z > 1.28, we reject the null hypothesis, accept the alternative hypothesis and say that there is sufficient evidence at the α = 0.10 level of significance to conclude that the true mean heart rate during laughter exceeds 71 beats per minute.
Hope this Helps!!!
Please answer this correctly
Answer:
It’s a 1/2 chance it’s heads.
Step-by-step explanation:
Because there’s two sides
Answer:
1/2
Step-by-step explanation:
The probability of getting heads is 1 out of 2.
1/2
When you flip a coin, you either get heads or tails.
Find f(2) if f(x) = (x + 1)2
Answer:
9
Step-by-step explanation:
f(x) = (x + 1)^2
Let x=2
f(2) = (2 + 1)^2
= 3^2
= 9
PLEASE HELP***
solve for x
Answer:
x =8
Step-by-step explanation:
The angles are vertical angles which means they are equal
4x+112 = 9x+72
Subtract 4x from each side
112 = 5x+72
Subtract 72 from each side
40 = 5x
Divide each side by 5
40/5 =x
8 =x
Answer: [tex]x=8[/tex]
Step-by-step explanation: Because they are opposite angles, they can be assumed to be equal to each other. Therefore, you can set the two equations equal to each other, creating [tex]9x+72=4x+112[/tex], and from there you can subtract [tex]4x[/tex] as well as [tex]72[/tex] from both sides to get [tex]5x=40[/tex]. Then, divide both sides by [tex]5[/tex] to get [tex]x=8[/tex].
How do you solve this problem
Answer: Undefined
Step-by-step explanation:
For this problem, we know that 3ˣ=-9. All we have to do is figure out what x is.
We know that any integer raised to the power cannot be negative. The closest answer we can get is x=2 because 3²=9. Unfortunately, we are looking for -9. Therefore, this x is undefined.
Which answer shows 2.13786 times 10 Superscript 4 written in standard form?
Answer:
Answer:
21378.6
Step-by-step explanation:
You move the decimal point four places towards the right.
Answer:
21378.6
Step-by-step explanation:
What are the solution(s) to the quadratic equation 50 - x² = 0?
O x = 425
0 x = +675
x = 5/2
no real solution
Answer:
The answer is C.
Step-by-step explanation:
[tex]50-x^2=0[/tex]
[tex]x^2=50[/tex]
[tex]x=\pm \sqrt{50} =\pm \sqrt{25*2}=\pm 5\sqrt{2}[/tex]
The answer is C (I am assuming that it isn't 5/2).
Does this sample have a bias
A concerned citizen
stands outside of the
courthouse with a petition to
outlaw tickets being given for
parking meters to be
removed.
Answer:
Yes, the sample has a bias
Step-by-step explanation:
Bias is the term used in statistics to describe a systematic distortion in the samples obtained for the parameter being estimated. It is evident by obtaining values higher or lower than that of the average population for the parameter being measured. As such the data is a misrepresentation of the population and cannot be trusted to give a good indices of things.
This sample has a bias because the concerned citizen opted to use a convenience sampling instead of using random sampling. In random sampling, every individual has an equal chance of being chosen which is unlike the convenience sampling when only a specific group of individuals can be chosen. In this case, the bias was introduced when the citizen went to stand outside the courthouse with his petition, the citizen should have taken samples across diverse geographical locations and professional institutions. The implication of this sampling is that a significant percentage of the population has been sidelined (considering they would not be in or around the courthouse) from the sampling and the sampling has been restricted to only those who have business around the courthouse. The result is that whatever samples he/she obtains will not be an accurate representation of the parameter being measure from the population.
As such, the sampling technique is biased
Gravel is being dumped from a conveyor belt at a rate of 35 ft3/min, and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 9 ft high? (Round your answer to two decimal places.)
Answer:
dh/dt ≈ 0.55 ft/min
Step-by-step explanation:
The volume is given by the formula ...
V = (1/3)πr²h
We have r = h/2, so the volume as a function of height is ...
V = (1/3)π(h/2)²h = (π/12)h³
Then the rates of change are related by ...
dV/dt = (π/4)h²·dh/dt
dh/dt = (4·dV/dt)/(πh²) = 4(35 ft³/min)/(π(9 ft)²)
dh/dt ≈ 0.55 ft/min
Solve the inequality and graph the solution set?
Answer:
0 < x < 8
Step-by-step explanation:
| x-4| < 4
There are two solutions one positive and one negative. Remember to flip the inequality on the negative solution
x-4 <4 and x-4 > -4
Add 4 to each side
x -4+4 <4+4 and x-4+4 > -4+4
x < 8 and x > 0
0 < x < 8
open circles at 0 and 8 and a line connecting them
Which represents two rays that intersect at a common endpoint
An angle is the intersection of two noncollinear rays at a common endpoint. The rays are called sides and the common endpoint is called the vertex.
this graph shows the outside temperature (in degrees celsius) over the course of 12 hours, starting at midnight (x=0)
Answer:
You can add graph by using the edit button and uploading a picture of the screen. if you then cut the picture before uploading so it just shows the graphand not the question. We cna then try answer for you.
Step-by-step explanation:
Generally graphs starting at point x=0 would show a different value for y by looking and counting up to its temperature.
if this shows positive it would be above the x axis line if it shows negative it would be a minus value below the x axis line under zero on y.
Therefore when we get to hr 2 and see this change you can count across and count up and see the rate of change is either 1,2,3,4,56,7,8 etc difference or 1x 2x 3x 4 x 5x 6 x as multiples.This then indicates a scale change at certain points.
I need help to solve this grid question
Answer:
20 x 20 grid = 1066.67 cm wire
Step-by-step explanation:
Using unitary method
9 grid = 24 cm wire
1 grid = [tex]\frac{24}{9}[/tex] cm wire
Multiplying both sides by 400 (20 x 20)
400 grid = [tex]\frac{24}{9} * 400[/tex] cm
20 x 20 grid = 1066.67 cm wire
The waiting time for a train has a uniform distribution between 0 and 10 minutes. What is the probability that the waiting time for this train is more than 4 minutes on a given day? Answer: (Round to two decimal place.)
Answer:
0.6 = 60% probability that the waiting time for this train is more than 4 minutes on a given day
Step-by-step explanation:
An uniform probability is a case of probability in which each outcome is equally as likely.
For this situation, we have a lower limit of the distribution that we call a and an upper limit that we call b.
The probability that we find a value X higher than x is given by the following formula.
[tex]P(X > x) = \frac{b - x}{b-a}[/tex]
The waiting time for a train has a uniform distribution between 0 and 10 minutes.
This means that [tex]a = 0, b = 10[/tex]
What is the probability that the waiting time for this train is more than 4 minutes on a given day?
[tex]P(X > x) = \frac{b - x}{b-a}[/tex]
[tex]P(X > 4) = \frac{10 - 4}{10 - 0} = 0.6[/tex]
0.6 = 60% probability that the waiting time for this train is more than 4 minutes on a given day
Please answer this correctly
Answer:
the correct answer is
Step-by-step explanation:
So, the probability is:P(greater than 4)=26=13. This is a theoretical probability, which is the observed number of favorable outcomes out of a certain number of trials. For instance, suppose you rolled the six-sided die five times, and got the following results:2,6,4,5,6
hope this help you!!!!!
Answer:
1/5 chance.
Step-by-step explanation:
There is only one number, 5, that is greater than 4 and there are 5 total numbers so there is a 1/5 chance selecting that number.
You randomly select one card from a 52-card deck. Find the probability of selecting the four of spades or the ace of clubs.
(Type answer an integer or a fraction. Simplify your answer.)
Answer:
1/26
Step-by-step explanation:
There is 1 four of spades, and 1 ace of clubs.
So the probability is 2/52, or 1/26.
Answer:
P(four of spades or ace of clubs)= 1/26
Step-by-step explanation:
In a deck of 52 cards, there is one four of spaces and one ace of clubs. we want to find the probability of selecting those cards.
P(four of spades or ace of clubs)=four of spades+ace of clubs/total cards
There is 1 four of spades and 1 ace of clubs. 1+1=2
P(four of spades or ace of clubs)=2/total cards
There are 52 total cards in a deck.
P(four of spades or ace of clubs)=2/52
This fraction can be simplified. Both the numerator (top number) and denominator (bottom number) can be divided by 2.
P(four of spades or ace of clubs)= (2/2) / (52/2)
P(four of spades or ace of clubs)= 1/26
A child takes a bouquet of flowers and a group of small vases, and places a single flower in each vase, so
that no vases or flowers are left over. What mathematical relationship have they just established between
the bouquet of flowers and the group of vases?
Answer:
Step-by-step explanation:
Determine the total number of roots of each polynomial function. g(x) = 5x - 12x2 + 3
Answer:
2 total roots
x = -1/3, 3/4
Step-by-step explanation:
We can use the discriminant b² - 4ac to find how many roots a polynomial has.
Answer:
2Step-by-step explanation:
Edginuity 2021
Can someone explain to me? i don't understand it
Step-by-step explanation:
I will do 12 and 14 as examples.
12) Angles of a triangle add up to 180°.
m∠P + m∠Q + m∠R = 180
5x − 14 + x − 5 + 2x − 9 = 180
8x − 28 = 180
8x = 208
x = 26
m∠P = 5x − 14 = 116
m∠Q = x − 5 = 21
m∠R = 2x − 9 = 43
14) If two sides of a triangle are equal, then the angles opposite those sides are also equal.
(Conversely, if two angles are equal, then the sides opposite those angles are also equal. Such a triangle is called an isosceles triangle.)
BC ≅ BD, so m∠C = m∠D.
5x − 19 = 2x + 14
3x = 33
x = 11
m∠B = 13x − 35 = 108
m∠C = 5x − 19 = 36
m∠D = 2x + 14 = 36
Suppose the Acme Drug Company develops a new drug, designed to prevent colds. The company states that the drug is equally effective for men and women. To test this claim, they choose a simple random sample of 100 women and 200 men from a population of 100,000 volunteers. At the end of the study, 38% of the women caught a cold; and 51% of the men caught a cold. Based on these findings, can we reject the company's claim that the drug is equally effective for men and women? Use a 0.05 level of significance.
a) Hypothesis: (In both symbols and words)
b) Decision rule:
c) Calculation:
d) Conclusion:
Answer:
a) Null hypothesis: the drug is equally effective for men and women (company's claim)
Alternative hypothesis: the drug effectiveness significantly differs for men and women.
[tex]H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\neq 0[/tex]
b) If the P-value is smaller than the significance level, the null hypothesis is rejected. If not, the null hypothesis failed to be rejected.
c) In the explanation.
d) As the P-value (0.0342) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that the drug effectiveness significantly differs for men and women.
The company's claim is rejected.
Step-by-step explanation:
This is a hypothesis test for the difference between proportions.
The claim of the company, will be stated in the null hypothesis. We will test if there is evidence against that claim to reject it or not.
Then, the test claim is that the drug effectiveness significantly differs for men and women.
The null and alternative hypothesis are:
[tex]H_0: \pi_1-\pi_2=0\\\\H_a:\pi_1-\pi_2\neq 0[/tex]
The significance level is 0.05.
The sample 1 (women), of size n1=100 has a proportion of p1=0.38.
The sample 2 (men), of size n2=200 has a proportion of p2=0.51.
The difference between proportions is (p1-p2)=-0.13.
[tex]p_d=p_1-p_2=0.38-0.51=-0.13[/tex]
The pooled proportion, needed to calculate the standard error, is:
[tex]p=\dfrac{X_1+X_2}{n_1+n_2}=\dfrac{38+102}{100+200}=\dfrac{140}{300}=0.467[/tex]
The estimated standard error of the difference between means is computed using the formula:
[tex]s_{p1-p2}=\sqrt{\dfrac{p(1-p)}{n_1}+\dfrac{p(1-p)}{n_2}}=\sqrt{\dfrac{0.467*0.533}{100}+\dfrac{0.467*0.533}{200}}\\\\\\s_{p1-p2}=\sqrt{0.002489+0.001244}=\sqrt{0.003733}=0.061[/tex]
Then, we can calculate the z-statistic as:
[tex]z=\dfrac{p_d-(\pi_1-\pi_2)}{s_{p1-p2}}=\dfrac{-0.13-0}{0.061}=\dfrac{-0.13}{0.061}=-2.1276[/tex]
This test is a two-tailed test, so the P-value for this test is calculated as (using a z-table):
[tex]P-value=2\cdot P(z<-2.1276)=0.0342[/tex]
As the P-value (0.0342) is smaller than the significance level (0.05), the effect is significant.
The null hypothesis is rejected.
There is enough evidence to support the claim that the drug effectiveness significantly differs for men and women.