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fomenos
3 years ago
10

Hi please write what's in the comments as the answers! I want people to know the answers!! Tai Chi is often recommended as a way

to improve balance and flexibility in the elderly. Below are
before-and-after flexibility ratings (on a 1 to 10 scale, 10 being most flexible) for a random sample of 8 men in their 80's who took Tai Chi lessons for six months.


Subject A B C D E F G H


Flexibility rating after Tai Chi 2 4 3 3 3 4 5 10


Flexibility rating before Tai Chi 1 2 1 2 1 4 2 10


(a) Explain why these are paired data.


(b) Calculate and interpret the mean difference.


(c) Researchers would like to know if the true mean difference (after-before) in flexibility rating for men in their 80's who take Tai Chi lessons for 6 months is greater than zero. The P-value of this test is 0.004. Interpret this value.


(d) If the result of this study is statistically significant, can you conclude that the difference in the mean flexibility rating was caused by the Tai Chi lessons? Why or why not?
Mathematics
1 answer:
Helen [10]3 years ago
5 0

Answer:

a) Because same men were sampled before and after Tai Chi

b) d = 1.375

c) If we assume that the mean difference is not greater than zero, the probability of obtaining the mean difference equal to 1.375 is 0.004

d) Yes.

Step-by-step explanation:

Part a)

We are given the data of 8 men and their flexibility rating before and after practicing Tai Chi Lessons. This data is an example of paired data because the same men were sampled before and after practicing Tai Chi for this study. Whenever the subjects of the study remain same, we call it a paired data and use paired data t-test for carrying out the hypothesis testing.

Part b) Mean Difference

We have to calculate the difference in flexibility rating of each man, and then find the mean of those differences. This mean would be termed as the mean difference.

The difference we will calculate is After - Before, as said in the question.

Subject A: 2 - 1 = 1

Subject B: 4 - 2 = 2

Subject C: 3 - 1 = 2

Subject D: 3 - 2 = 1

Subject E: 3 - 1 = 2

Subject F: 4 - 4 = 0

Subject G: 5 - 2 = 3

Subject H: 10 - 10 = 0

Mean of these differences would be sum of these differences divided by number of sums which is 8.

So, mean difference = 11/8 = 1.375

Interpretation: Since the mean difference is greater than 0, it seems that Tai Chi improved the flexibility rating of these 8 men in general.

Part c)

Researchers want to test is the mean value is greater than 0. So, the null and alternate hypothesis would be:

Null Hypothesis: d ≤ 0

Alternate Hypothesis: d > 0

Here, d stands for the mean difference (after -before)

The p-value of this hypothesis test came out to be 0.004.

p-value is the probability of obtaining the result atleast as extreme as the actual test result, assuming the Null Hypothesis to be true. So, we can say that:

If we assume that the mean difference is not greater than zero, the probability of obtaining the mean difference equal to 1.375 is 0.004, which indeed is a very low value. Such a low p-value does not favor the Null Hypothesis.

Part d)

The p-value of this hypothesis test came out to be 0.004, as mentioned in the question. Since, the p-value is very very small, close to zero, we can reject the Null Hypothesis.

Conclusion: We have enough confidence to support the claim that the true mean difference(after - before) in flexibility rating for men in their 80's who take Tai Chi is greater than Zero. This means, Tai Chi helps in increasing the flexibility rating.

So, Yes we can conclude that the difference in the mean flexibility rating was caused by the Tai Chi lessons

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3 years ago
The mean number of words per minute (WPM) read by sixth graders is 8888 with a standard deviation of 1414 WPM. If 137137 sixth g
Bingel [31]

Noticing that there is a pattern of repetition in the question (the numbers are repeated twice), we are assuming that the mean number of words per minute is 88, the standard deviation is of 14 WPM, as well as the number of sixth graders is 137, and that there is a need to estimate the probability that the sample mean would be greater than 89.87.

Answer:

"The probability that the sample mean would be greater than 89.87 WPM" is about \\ P(z>1.56) = 0.0594.

Step-by-step explanation:

This is a problem of the <em>distribution of sample means</em>. Roughly speaking, we have the probability distribution of samples obtained from the same population. Each sample mean is an estimation of the population mean, and we know that this distribution behaves <em>normally</em> for samples sizes equal or greater than 30 \\ n \geq 30. Mathematically

\\ \overline{X} \sim N(\mu, \frac{\sigma}{\sqrt{n}}) [1]

In words, the latter distribution has a mean that equals the population mean, and a standard deviation that also equals the population standard deviation divided by the square root of the sample size.

Moreover, we know that the variable Z follows a <em>normal standard distribution</em>, i.e., a normal distribution that has a population mean \\ \mu = 0 and a population standard deviation \\ \sigma = 1.

\\ Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}} [2]

From the question, we know that

  • The population mean is \\ \mu = 88 WPM
  • The population standard deviation is \\ \sigma = 14 WPM

We also know the size of the sample for this case: \\ n = 137 sixth graders.

We need to estimate the probability that a sample mean being greater than \\ \overline{X} = 89.87 WPM in the <em>distribution of sample means</em>. We can use the formula [2] to find this question.

The probability that the sample mean would be greater than 89.87 WPM

\\ Z = \frac{\overline{X} - \mu}{\frac{\sigma}{\sqrt{n}}}

\\ Z = \frac{89.87 - 88}{\frac{14}{\sqrt{137}}}

\\ Z = \frac{1.87}{\frac{14}{\sqrt{137}}}

\\ Z = 1.5634 \approx 1.56

This is a <em>standardized value </em> and it tells us that the sample with mean 89.87 is 1.56<em> standard deviations</em> <em>above</em> the mean of the sampling distribution.

We can consult the probability of P(z<1.56) in any <em>cumulative</em> <em>standard normal table</em> available in Statistics books or on the Internet. Of course, this probability is the same that \\ P(\overline{X} < 89.87). Then

\\ P(z

However, we are looking for P(z>1.56), which is the <em>complement probability</em> of the previous probability. Therefore

\\ P(z>1.56) = 1 - P(z

\\ P(z>1.56) = P(\overline{X}>89.87) = 0.0594

Thus, "The probability that the sample mean would be greater than 89.87 WPM" is about \\ P(z>1.56) = 0.0594.

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weeeeeb [17]

Answer:

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Step-by-step explanation:

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anzhelika [568]
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Explanation:

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Since sine is negative, so is cosecant as this is the reciprocal of sine

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In quadrant IV, cosine is positive as x > 0 here. So the ratio tan = sin/cos is going to be negative. We have a negative over a positive when we divide.

Because tangent is negative, so is cotangent.

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Answer:

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Step-by-step explanation:

Rule: an even number of - signs written one after another produces a plus number.

So - - 3 = 3

The rule also lets you do something strange like

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Rule: an odd number of minus signs produces a minus number.

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x + 3 - 3 = 10 - 3      

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2 years ago
Read 2 more answers
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