Final answer:
Using Chebyshev's Rule, at least 857 out of 875 cars are expected to have prices between $26700 and $46300, and at least 850 out of 875 cars between $28100 and $44900.
Step-by-step explanation:
To find out at least how many new car prices lie within certain ranges using Chebyshev's Rule, we need to calculate the number of standard deviations the given range covers from the mean. For the first case, the range is $26700 to $46300 around a mean of $36500. We begin by determining how many standard deviations $26700 and $46300 are from the mean:
- Lower deviation: (36500 - 26700) / 1400 = 7 standard deviations below the mean.
- Upper deviation: (46300 - 36500) / 1400 = 7 standard deviations above the mean.
According to Chebyshev's Rule, the proportion of values within k standard deviations of the mean (for k > 1) is at least 1 - 1/k2. Here, k = 7, so the proportion is at least 1 - 1/49, which is approximately 1 - 0.0204, or 0.9796.
Thus, for a sample of 875 cars, at least 875 * 0.9796 ≈ 857 cars would lie between $26700 and $46300.
For the range of $28100 to $44900, we calculate similarly:
- Lower deviation: (36500 - 28100) / 1400 = 6 standard deviations below the mean.
- Upper deviation: (44900 - 36500) / 1400 = 6 standard deviations above the mean.
Applying Chebyshev's Rule with k = 6, the proportion of values is at least 1 - 1/36, or approximately 0.9722.
For a sample of 875 cars, at least 875 * 0.9722 ≈ 850 cars would lie between $28100 and $44900.