Final answer:
To find the percentage of prices between $50 and $72 given the normal distribution, we calculate the Z-scores for each price and then subtract the corresponding probabilities. After doing the math, we find the percentage to be 19.7% when rounded to the nearest tenth.
Step-by-step explanation:
To find the percentage of prices between $50 and $72 when the prices are normally distributed with a mean of $37.31 and a variance of $34.722, we first need to convert the prices into Z-scores. The formula for the Z-score is Z = (X - mean) / standard deviation. First, we calculate the standard deviation, which is the square root of the variance, thus standard deviation = $34.72.
Next, we find the Z-scores for $50 and $72:
- Z for $50 = ($50 - $37.31) / $34.72 = 0.3646
- Z for $72 = ($72 - $37.31) / $34.72 = 0.9983
We then consult the standard normal distribution table to find the probabilities associated with these Z-scores:
- P(Z < 0.3646) ≈ 0.6423 (64.23%)
- P(Z < 0.9983) ≈ 0.8389 (83.89%)
Now, to find the percentage of prices between $50 and $72, we subtract the lower probability from the higher probability:
Percentage = P(Z < 0.9983) - P(Z < 0.3646) = 83.89% - 64.23% = 19.66%
Rounded to the nearest tenth, the answer is 19.7%, which is not listed in the options provided. There may have been a typo or mistake in the options given. The closest option to our calculated value is 19.8%, labeled as option A.