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astra-53 [7]
3 years ago
10

Given: m angle P equals open parentheses 5 x minus 27 close parentheses degree, m angle Q equals open parentheses 4 x minus 7 cl

ose parentheses degree, and m angle R equals open parentheses 3 x plus 10 close parentheses degree. Which side in increment P Q R is the shortest?
Mathematics
1 answer:
max2010maxim [7]3 years ago
5 0

Answer: Side QR

Step-by-step explanation: The sum of a triangle internal angles is 180°. So to determine the angles, we calculate x:

5x - 27  +4x - 7 + 3x + 10 = 180

12x = 204

x = 17

With x, solve for the angles:

m∠P = 5(17) - 27 = 58

m∠Q = 4(17) - 7 = 61

m∠R = 3(17) + 10 = 61

The relationship between angles and sides in any triangle:

  • the largest angle is opposite the largest side;
  • the middle angle is opposite to the middle-sized side;
  • the shortest angle and side are opposite to each other;

So, the shortest side of PQR is the side opposite to the shortest angle.

The shortest angle is m∠P = 58°. Therefore, the shortest side is QR

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According to statistician Persi Diaconis, the probability of a penny landing heads when it is spun on its edge is only about 0.2
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Answer:

a) The p-value obtained from the one-proportion applet is more valid because a z-test statistic shouldn't have been used for the other obtained p-value. Check Explanation for more Explanation.

b) No, there isn't enough evidence to suggest that a spun penny will land heads more that 20% of the time in the long run.

Step-by-step explanation:

The p-value for this problem was obtained from a one proportion simulation applet and another obtained using a one proportion z-test p-value.

But one of the conditions for the use of the z-test or the z-distribution in obtaining the p-value is that information on the population mean and standard deviation should be known or the sample size should be large enough such that the properties of the sample should approximate the properties of the the population distribution.

But for this question and hypothesis test, the sample that the we are working with is only of sample size 12 with no information on the population standard deviation provided, hence, the p-value obtained from the z-test statistic one proportion test is not a valid enough one due to this reason.

Plus, on calculating this p-value manually, it was obtained to be 0.078, to justify this explanation as it is very close to.the value obtained using the simulation applet.

Manual way of calculating

t = (x - μ)/σₓ

x = 5/12 = 0.41667

μ = p₀ = 0.20

σₓ = standard error = √[p(1-p)/n]

where n = Sample size = 12

σₓ = √[0.4167×0.5833/12] = 0.1423

t = (0.4167 - 0.20) ÷ 0.1423

t = 1.52

checking the tables for the p-value of this t-statistic

Degree of freedom = df = n - 1 = 12 - 1 = 11

Significance level = 0.05 (This is used when no significance level is provided in the question)

The hypothesis test uses a one-tailed condition because we're testing only in one direction.

p-value (for t = 1.52, at 0.05 significance level, df = 11, with a one tailed condition) = 0.07836

b) To know which conclusion to draw, we need to first define 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.

For this question, the null hypothesis is that there isn't enough evidence to suggest that a spun penny will land heads more that 20% of the time in the long run.

And the alternative hypothesis is that there is enough evidence to suggest that a spun penny will land heads more that 20% of the time in the long run.

Mathematically, if p is the proportion of times the spun penny will turn up heads in the long run,

The null hypothesis is represented as

H₀: p ≤ 0.20

The alternative hypothesis is represented as

Hₐ: p > 0.20

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 (usually used when the significance level for the test isn't specified)

p-value = 0.0770

0.0770 > 0.05

Hence,

p-value > significance level

This means that we fail to reject the null hypothesis & say there isn't enough evidence to suggest that a spun penny will land heads more that 20% of the time in the long run.

Hope this Helps!!!

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