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 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, you win $35; otherwise you lose $1. Complete parts (a) through (g) below.What are the mean and standard deviation of the sampling distribution of x overbarx? Round your results to the nearest penny.
To find the mean (µ) and standard deviation (σ) of the sampling distribution of x̄ (the sample mean), we need to use the formulas:
µ = E(x̄) = μ (population mean)
σ = σ / sqrt(n) (population standard deviation divided by the square root of the sample size)
In this case, the population consists of the 38 slots numbered 0, 00, 1, 2,..., 36.
a) The population mean (μ) is the average number on the wheel. To calculate it, we sum up all the numbers on the wheel and divide by the total number of slots:
μ = (0 + 00 + 1 + 2 + ... + 36) / 38
You can calculate this sum and then divide by 38 to find the population mean.
b) We need to determine the population standard deviation (σ) to calculate the standard deviation of the sampling distribution. This involves calculating the variance first. To find the variance, we need to calculate the squared deviation of each number from the population mean and then find the average of these squared deviations. Finally, taking the square root of the variance gives us the standard deviation.
First, calculate the squared deviation for each number on the wheel as follows:
For each number, calculate: (number - population mean)^2
Sum up all these squared deviations and divide by the total number of slots (38) to find the variance.
Finally, take the square root of the variance to obtain the standard deviation.
c) Once we have the population standard deviation (σ), we can calculate the standard deviation of the sampling distribution (σx̄) using the formula:
σx̄ = σ / sqrt(n)
Note that in this problem, the sample size (n) is not given, so it's not possible to calculate the exact value of σx̄. However, you can calculate it once the value of n is provided.
d) If you have the value of σx̄ and want to round it to the nearest penny, simply round the value to two decimal places.
In summary:
a) Calculate the population mean (μ) using the sum of all the numbers on the wheel divided by 38.
b) Calculate the variance by calculating the squared deviation for each number on the wheel, summing them up, and dividing by 38. Then, take the square root of the variance to find the standard deviation (σ).
c) Once the sample size (n) is given, calculate the standard deviation of the sampling distribution (σx̄) using the formula σx̄ = σ / sqrt(n).
d) Round the calculated σx̄ to the nearest penny by rounding it to two decimal places.
To calculate the mean and standard deviation of the sampling distribution of the sample mean (x̄) in this scenario, we need to use the formulas:
Mean of the sampling distribution (µx̄) = µ = Population mean
Standard deviation of the sampling distribution (σx̄) = σ / √n
where:
- µ represents the population mean
- σ represents the population standard deviation
- n represents the sample size
In this particular scenario, the population mean (µ) would be the expected value of a single trial of playing roulette, which can be calculated as follows:
µ = (P(win) * winnings for winning) + (P(lose) * winnings for losing)
= (1/38 * $35) + (37/38 * -$1)
= ($35/38) - ($37/38)
= -$2/38
≈ -$0.053
So, the population mean is approximately -$0.053.
The population standard deviation (σ) for playing roulette can be calculated using the formula for standard deviation:
σ = √[(Sum of (x - µ)^2) / N]
where:
- N represents the population size
- x represents each value in the population
In this case, N = 38 (number of slots on the roulette wheel), and x would be the values for each slot, which are -1, 0, 1, 2,..., 36. So we can calculate the population standard deviation as follows:
σ = √[(Sum of (x - µ)^2) / N]
= √[(Sum of (x - (-$0.053))^2) / 38]
To calculate standard deviation, we will need to find the sum of the squares of the differences between each slot and the population mean. Here's how you can calculate it step-by-step:
1. Subtract the population mean from each slot value:
-1 - (-$0.053), 0 - (-$0.053), 1 - (-$0.053), 2 - (-$0.053),..., 36 - (-$0.053)
2. Square each of the differences obtained in Step 1:
(-1 - (-$0.053))^2, (0 - (-$0.053))^2, (1 - (-$0.053))^2, (2 - (-$0.053))^2,..., (36 - (-$0.053))^2
3. Sum up all the squared differences obtained in Step 2.
4. Divide the sum from Step 3 by the population size (N = 38).
5. Take the square root of the result obtained in Step 4 to find the population standard deviation (σ).
After performing these calculations, you will have the value for the standard deviation (σ) of the population.
Once you have the population mean (µ) and population standard deviation (σ), the mean (µx̄) and standard deviation (σx̄) of the sampling distribution of the sample mean (x̄) can be determined using the formulas mentioned previously.
I hope this helps! Let me know if you need any further assistance.