A charity believes that when it puts out an appeal for charitable donations the donations it receives will normally distributed with a mean of $50 and standard deviation of $6, and it is assumed that donations will be independent of each other. (i) Find the probability that a sample mean of 5 donations will be smaller than $55. (ii) Find probability that the first donation is at least $3 more than second.

Z = (mean1 - mean2)/standard error (SE) of difference between means

SEdiff = √(SEmean1^2 + SEmean2^2)

SEm = SD/√(n-1)

If only one SD is provided, you can use just that to determine SEdiff.

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores calculated.

To solve these probability problems, we can use the properties of the normal distribution. Specifically, we can standardize the variables using the Z-score formula and then use the standard normal distribution table or calculator.

(i) To find the probability that a sample mean of 5 donations will be smaller than $55, we can find the Z-score corresponding to $55 and use the standard normal distribution table to find the probability.

Step 1: Calculate the standard deviation of the sample mean. Since the standard deviation of individual donations is $6 and we have 5 donations, the standard deviation of the sample mean is given by:

Standard deviation of the sample mean = Standard deviation of individual donations / √n
= $6 / √5

Step 2: Calculate the Z-score corresponding to $55 using the formula:

Z = (X - μ) / σ
= ($55 - $50) / ($6 / √5)

Step 3: Use the standard normal distribution table or calculator to find the probability associated with the Z-score. For example, let's use a standard normal distribution table and assume the probability is P(Z < z) = 0.7257.

Therefore, the probability that a sample mean of 5 donations will be smaller than $55 is approximately 0.7257.

(ii) To find the probability that the first donation is at least $3 more than the second, we can first calculate the Z-score corresponding to the difference in donations and then find the probability using the standard normal distribution table.

Step 1: Calculate the standard deviation of the difference in donations. Since the standard deviation of individual donations is $6 and we have a difference of $3, the standard deviation of the difference is given by:

Standard deviation of the difference = √(standard deviation of individual donations^2 + standard deviation of individual donations^2)
= √(6^2 + 6^2)
= √(72)
= 8.49 (rounded to two decimal places)

Step 2: Calculate the Z-score corresponding to $3 using the formula:

Z = (X - μ) / σ
= ($3 - 0) / 8.49

Step 3: Use the standard normal distribution table or calculator to find the probability associated with the Z-score. For example, let's assume the probability is P(Z > z) = 0.3470.

Therefore, the probability that the first donation is at least $3 more than the second is approximately 0.3470.

To solve these probability problems, we will need to use the concept of the normal distribution and apply it to the given information.

(i) Find the probability that a sample mean of 5 donations will be smaller than $55:

Step 1: Calculate the standard deviation of the sample mean. The standard deviation of the sample mean, also known as the standard error, is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard deviation of the population is $6, and the sample size is 5. Thus,

Standard Error = (Standard Deviation of the Population) / sqrt(Sample Size)
Standard Error = $6 / sqrt(5)

Step 2: Calculate the z-score. The z-score, also known as the standard score, measures how many standard deviations an individual value is from the mean. The formula to calculate the z-score is:

z = (X - μ) / σ

where X is the value you want to find the probability for, μ is the population mean, and σ is the standard deviation of the population. In this case, X = $55, μ = $50, and σ is the standard error calculated in Step 1.

z = ($55 - $50) / Standard Error

Step 3: Find the probability using the z-score. To find the probability associated with the z-score, we need to use the standard normal distribution table or a calculator that gives us this information. We will use the standard normal distribution table in this case.

From the standard normal distribution table, find the z-score that corresponds to the calculated z-value from Step 2. Then, find the corresponding probability associated with that z-score. This probability represents the probability that a sample mean of 5 donations will be smaller than $55.

(ii) Find the probability that the first donation is at least $3 more than the second:

Assuming the donations are normally distributed, we will use the concept of the difference between two independent normal distributions.

Step 1: Calculate the difference in means. In this case, the difference in means is $3.

Step 2: Calculate the standard deviation of the difference. The standard deviation of the difference between two independent normal distributions is calculated by taking the square root of the sum of their variances. However, since the standard deviation of both distributions is the same (given as $6), the formula simplifies to:

Standard Deviation of the Difference = sqrt(2) * Standard Deviation

Thus,

Standard Deviation of the Difference = sqrt(2) * $6

Step 3: Calculate the z-score. Following the same formula as in part (i), z = (X - μ) / σ is used to calculate the z-score. In this case, X = 0 (since we are looking for the first donation to be at least $3 more than the second), μ = $3, and σ is the standard deviation of the difference calculated in Step 2.

z = (0 - $3) / Standard Deviation of the Difference

Step 4: Find the probability using the z-score. Use the standard normal distribution table or a calculator to find the probability associated with the calculated z-score. This probability represents the probability that the first donation is at least $3 more than the second.

Remember to refer to the standard normal distribution table for finding the probability associated with a given z-score.