Final answer:
To find the IQR for the given distribution of children's weights, calculate the first and third quartiles using the z-scores for 25th and 75th percentiles and subtract them to obtain the IQR, which is approximately 15.52 lbs.
Step-by-step explanation:
The question asks for the interquartile range (IQR) for the weight of children at a certain age, whose weights follow a normal distribution with a mean of 70.5 lbs and a standard deviation of 11.5 lbs. The IQR is the difference between the third quartile (75th percentile) and the first quartile (25th percentile). For a normal distribution, these quartiles can be found using z-scores that correspond to the cumulative probabilities of 0.25 and 0.75, respectively. For a standard normal distribution, these z-scores are approximately -0.6745 for the first quartile and +0.6745 for the third quartile. To convert these into the weights for our specific distribution, we can use the following formulas:
- Q1 = mean + (z-score for 25th percentile * standard deviation)
- Q3 = mean + (z-score for 75th percentile * standard deviation)
Therefore:
- Q1 = 70.5 + (-0.6745 * 11.5) = 70.5 - 7.75675 ≈ 62.74 lbs
- Q3 = 70.5 + (0.6745 * 11.5) = 70.5 + 7.75675 ≈ 78.26 lbs
The IQR is then calculated as:
IQR = Q3 - Q1 = 78.26 lbs - 62.74 lbs ≈ 15.52 lbs