1. Suppose the distribution of birth weights of babies born at Big General Hospital is normal. If the mean weight is 7.5 lb. and the standard deviation is 1.25 lb, about 50% of the babies will weight between....
2. Joe runs a carnival game in which he charges $2 for a player to randomly select 3 cards from a 52 card deck. If the player gets 3 cards of the same color, the player is handed a $5.00 bill. What is Joe's expected profit per play? (Round to th nearest cent).
3. Suppose a random sample of size n=144 is obtained from a population with a population mean 75 and population standard deviation 10. Use the central limit theorem to approximate the probability P(72 < x bar < 78), where x bar denotes the sample mean.
Sorry, I'm not sure how to do these problems. Much thanks!
1. To find out what weight range 50% of the babies will fall between, we can use the properties of the normal distribution. Since the distribution of birth weights is normal, we can use the Z-score formula to standardize the weights and then use a Z-table to find the corresponding percentile.
The Z-score formula is: Z = (X - μ) / σ
Where:
Z is the standardized score
X is the individual value
μ is the mean of the distribution
σ is the standard deviation of the distribution
In this case:
X = 7.5 lb (mean weight)
μ = 7.5 lb (mean weight)
σ = 1.25 lb (standard deviation)
To find the Z-score for the lower percentile (25%), we subtract the mean from the X value and divide by the standard deviation:
Z1 = (X - μ) / σ = (7.5 - 7.5) / 1.25 = 0
Similarly, to find the Z-score for the upper percentile (75%), we subtract the mean from the X value and divide by the standard deviation:
Z2 = (X - μ) / σ = (7.5 - 7.5) / 1.25 = 0
Since the Z-scores are both zero, we can look up the corresponding percentile in the Z-table, which is 0.5000. This means that 50% of the babies will weigh between the mean weight (7.5 lb) and the mean weight (7.5 lb).
2. To find Joe's expected profit per play, we need to calculate the probability of winning and the amount of money won in each case.
There are two cases in which the player wins:
- Getting 3 cards of the same color (red or black)
- Getting 3 cards of the same suit (hearts, diamonds, clubs, or spades)
First, let's calculate the probability of getting 3 cards of the same color:
There are 26 red cards and 26 black cards in a standard 52-card deck. The probability of getting 3 cards of the same color can be calculated as follows:
P(getting 3 cards of the same color) = P(getting 3 red cards) + P(getting 3 black cards)
P(getting 3 red cards):
There are 26 red cards in the deck, and we need to choose 3 of them.
P(getting 1st red card) = 26/52
P(getting 2nd red card) = 25/51 (we have already taken 1 red card out)
P(getting 3rd red card) = 24/50 (we have already taken 2 red cards out)
P(getting 3 red cards) = (26/52) * (25/51) * (24/50)
Similarly, P(getting 3 black cards) will have the same probability.
Now, let's calculate the probability of getting 3 cards of the same suit (hearts, diamonds, clubs, or spades):
There are 4 suits in a standard deck. The probability of getting 3 cards of the same suit can be calculated as follows:
P(getting 3 cards of the same suit) = P(getting 3 hearts) + P(getting 3 diamonds) + P(getting 3 clubs) + P(getting 3 spades)
P(getting 3 hearts):
There are 13 hearts in the deck, and we need to choose 3 of them.
P(getting 3 hearts) = (13/52) * (12/51) * (11/50)
Similarly, P(getting 3 diamonds), P(getting 3 clubs), and P(getting 3 spades) will have the same probability.
Now, we can calculate the total probability of winning:
P(winning) = P(getting 3 cards of the same color) + P(getting 3 cards of the same suit)
Next, let's calculate the amount of money won in each case:
- If the player wins, they receive a $5 bill.
- If the player loses, Joe keeps the $2.
Finally, we can calculate Joe's expected profit per play:
Expected profit per play = (P(winning) * amount won) - (P(losing) * amount lost)
Expected profit per play = (P(winning) * $5) - (P(losing) * $2)
3. To approximate the probability P(72 < x̄ < 78), where x̄ denotes the sample mean, we can use the central limit theorem. The central limit theorem states that the distribution of sample means, for a large enough sample size, follows a normal distribution regardless of the shape of the original population distribution.
Given that the population mean (μ) is 75 and the population standard deviation (σ) is 10, we can apply the central limit theorem to approximate the probability.
To use the central limit theorem, we need to find the mean and standard deviation of the sampling distribution of the sample mean:
The mean of the sampling distribution of the sample mean (usually denoted as μx̄) is equal to the population mean (μ):
μx̄ = μ = 75
The standard deviation of the sampling distribution of the sample mean (usually denoted as σx̄) is calculated using the formula: σx̄ = σ / √n
Where:
σ is the population standard deviation
n is the sample size
Plugging in the values:
σx̄ = 10 / √144 = 10 / 12 = 0.8333 (approximately)
Now, we have transformed the original problem into finding the probability P(72 < x̄ < 78) in a standard normal distribution with a mean of 75 and a standard deviation of 0.8333.
To calculate this probability, we need to find the Z-scores for the lower and upper limits:
Z1 = (72 - 75) / 0.8333
Z2 = (78 - 75) / 0.8333
Using a Z-table, we can find the corresponding cumulative probabilities:
P(72 < x̄ < 78) = P(Z1 < Z < Z2)
After finding the probabilities for Z1 and Z2 from the Z-table, subtracting them will give us the desired probability.