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liubo4ka [24]
3 years ago
13

In the game of​ roulette, a wheel consists of 38 slots numbered​ 0, 00,​ 1, 2,..., 36. To play the​ game, a metal ball is spun a

round the wheel and is allowed to fall into one of the numbered slots. If the number of the slot the ball falls into matches the number you​ selected, you win​ $35; otherwise you lose​ $1. Determine the mean and standard deviation of the random variable X.
Mathematics
1 answer:
Oxana [17]3 years ago
4 0

Answer:

E(X)=-0.0526

Sd(X)=5.763

Step-by-step explanation:

In statistics and probability analysis, the expected value "is calculated by multiplying each of the possible outcomes by the likelihood each outcome will occur and then summing all of those values".

The variance of a random variable Var(X) is the expected value of the squared deviation from the mean of X, E(X).

And the standard deviation of a random variable X is just the square root of the variance.  

Let X be a random variable which denotes the money you may win or lose on each spin.

In the game of roulette, a wheel consists of 38 slots. To play the game, a metal ball is spun around the wheel and is allowed to fall into one of the numbered slots. If the number of the slot the ball falls into matches the number you selected, we win $35; otherwise we lose $1.  So we have just one possibility to win and 37 of lose on an individual game.

___________________________

X            P(X)

___________________________

35           1/38

-1            37/38

___________________________

In order to calculate the expected value we can use the following formula:

E(X)=\sum_{i=1}^n X_i P(X_i)

And if we use the values obtained we got:

E(X)=(35)*(\frac{1}{38})+(-1)(\frac{37}{38})=-0.0526

In order to find the standard deviation we need to find first the second moment, given by :

E(X^2)=\sum_{i=1}^n X^2_i P(X_i)

And using the formula we got:

E(X^2)=(35^2)*(\frac{1}{38})+((-1)^2)(\frac{37}{38})=33.211

Then we can find the variance with the following formula:

Var(X)=E(X^2)-[E(X)]^2 =33.211-(-0.0526)^2 =33.208

And then the standard deviation would be given by:

Sd(X)=\sqrt{Var(X)}=\sqrt{33.208}=5.763

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An animal shelter spends $3.00 per day to care for each cat and $7.00 per day to care for each dog. The animal shelter spent a t
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<h3>What is a system of equations?</h3>

A system having more than two equations is known as a system of equations.

Given that, the shelter spends $3 per day to care for each cat and $7 per day to care for each dog.

Let the number of cats be x and the number of dogs is y.

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A marketing manager samples 150 people and finds that 87 of them have made a purchase on the internet within the past month. a.
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Answer:

a) \hat p = \frac{87}{150}= 0.58

b) We assume for this case a confidence level of 95%

n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}   (b)  

And replacing into equation (b) the values from part a we got:

n=\frac{0.58(1-0.58)}{(\frac{0.03}{1.96})^2}=1039.793  

And rounded up we have that n=1040

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".  

The margin of error is the range of values below and above the sample statistic in a confidence interval.  

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".  

The population proportion have the following distribution

p \sim N(p,\sqrt{\frac{p(1-p)}{n}})

Solution to the problem

Part a

For this case the estimated proportion of people who have made a purchase on the internet within the past month is given by:

\hat p = \frac{87}{150}= 0.58

Part b

We assume for this case a confidence level of 95%

In order to find the critical value we need to take in count that we are finding the interval for a proportion, so on this case we need to use the z distribution. Since our interval is at 95% of confidence, our significance level would be given by \alpha=1-0.95=0.05 and \alpha/2 =0.025. And the critical value would be given by:

t_{\alpha/2}=-1.96, t_{1-\alpha/2}=1.96

The margin of error for the proportion interval is given by this formula:  

ME=z_{\alpha/2}\sqrt{\frac{\hat p (1-\hat p)}{n}}    (a)  

And on this case we have that ME =\pm 0.03 and we are interested in order to find the value of n, if we solve n from equation (a) we got:  

n=\frac{\hat p (1-\hat p)}{(\frac{ME}{z})^2}   (b)  

And replacing into equation (b) the values from part a we got:

n=\frac{0.58(1-0.58)}{(\frac{0.03}{1.96})^2}=1039.793  

And rounded up we have that n=1040

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3 years ago
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