The formula to find the interquartile range (IQR) is:
IQR = Q3 - Q1
where Q1 and Q3 are the first and third quartiles, respectively. Since the distribution of weights of pumpkins is approximately normal, we can use the formula for the standard error of the mean to find the standard deviation of the sampling distribution of the mean:
SEM = σ / sqrt(n)
where σ is the population standard deviation, n is the sample size, and sqrt represents the square root function.
Plugging in the given values, we get:
SEM = 1.68 / sqrt(15)
≈ 0.4334
To find the first and third quartiles, we need to find the z-scores corresponding to the 25th and 75th percentiles of the standard normal distribution, respectively. These z-scores can be found using a standard normal distribution table or a calculator.
Using a standard normal distribution table, we find that:
- The z-score corresponding to the 25th percentile is approximately -0.6745.
- The z-score corresponding to the 75th percentile is approximately 0.6745.
Therefore, the first and third quartiles are:
Q1 = μ + z1 * SEM
= 8.3 - 0.6745 * 0.4334
≈ 7.998
Q3 = μ + z3 * SEM
= 8.3 + 0.6745 * 0.4334
≈ 8.602
Finally, we can calculate the interquartile range (IQR) as:
IQR = Q3 - Q1
≈ 8.602 - 7.998
≈ 0.604
Rounding to the nearest thousandth, we get:
IQR ≈ 0.604
Therefore, the value of the interquartile range (IQR) for the mean weight of 15 pumpkins is 0.604 pounds.