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astra-53 [7]
2 years ago
8

David's chance of making a free throw is 80%. In a game, he takes 10 free throws. What is the probability he makes 8 or fewer fr

ee throws out of the 10? Please leave your answer in combinations, products powers and sums. Please try to make your expression reasonably succinct.
Mathematics
1 answer:
maks197457 [2]2 years ago
4 0

Answer:

P(X\leq 8)= 1- P(X>8) = 1-P(X\geq 9)= 1-[P(X=9)+P(X=10)]

And we can find the individual probabilities like this:

P(X=9)=(10C9)(0.8)^{9} (1-0.8)^{10-9}

P(X=10)=(10C10)(0.8)^{10} (1-0.8)^{10-10}

And replacing we got:

P(X\leq 8)=1-[10(0.8)^1 (1-0.8)^{1} + (0.8)^{10}]

Step-by-step explanation:

Previous concepts

A Bernoulli trial is "a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted". And this experiment is a particular case of the binomial experiment.

The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".

The probability mass function for the Binomial distribution is given as:  

P(X)=(nCx)(p)^x (1-p)^{n-x}  

Where (nCx) means combinatory and it's given by this formula:  

nCx=\frac{n!}{(n-x)! x!}  

The complement rule is a theorem that provides a connection between the probability of an event and the probability of the complement of the event. Lat A the event of interest and A' the complement. The rule is defined by: P(A)+P(A') =1

Solution to the problem

For this case we want this probability:

P(X\leq 8)

And for this case we can use the comlement rule and we got:

P(X\leq 8)= 1- P(X>8) = 1-P(X\geq 9)= 1-[P(X=9)+P(X=10)]

And we can find the individual probabilities like this:

P(X=9)=(10C9)(0.8)^9 (1-0.8)^{10-9}

P(X=10)=(10C10)(0.8)^{10} (1-0.8)^{10-10}

And replacing we got:

P(X\leq 8)=1-[10(0.8)^9 (1-0.8)^{1} + (0.8)^{10}]

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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.

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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!!!

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