Final answer:
To find the probability that the mean of a sample will be between $70 and $80, we can use the Central Limit Theorem. The probability is approximately 0.9574.
Step-by-step explanation:
To find the probability that the mean of the sample will be between $70 and $80, we can use the Central Limit Theorem. Since the sample size is large (n = 64), we can assume that the distribution of the sample mean will be approximately normal. The mean of the sample will also be normally distributed with a mean of $75 and a standard deviation of $15/sqrt(64) = $1.875. To find the probability that the mean of the sample will be between $70 and $80, we can find the z-scores of $70 and $80 using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. Then, we can find the probability using the z-scores and the standard normal distribution table.
First, we calculate the z-score for $70:
z = ($70 - $75) / $1.875 = -2.67
Next, we calculate the z-score for $80:
z = ($80 - $75) / $1.875 = 2.67
Using the standard normal distribution table or a calculator, we can find the area between z = -2.67 and z = 2.67. The area between these two z-scores represents the probability that the mean of the sample will be between $70 and $80. Based on the table or calculator, the probability is approximately 0.9574.
Therefore, the correct answer is not provided in the options. The correct probability is approximately 0.9574, which is not listed as an option.