A company produces steel rods. The lengths of the steel rods are normally distributed with a mean of $119-\mathrm{cm}$ and a standard deviation of $0.5-\mathrm{cm}$. For shipment, 40 steel rods are bundled together. Round all answers to four decimal places if necessary.
a. What is the distribution of $X ? X \sim N$
c. For a single randomly selected steel $119 . \mathrm{cm}$.
d. For a bundled of 40 rods, find the probability that the average length is between $118.9-\mathrm{cm}$ and $119-\mathrm{cm}$.
e. For part d), is the assumption of normal necessary? Yeso No $0^{s}$
Answer
Final Answer: The probability that the average length of a bundle of 40 rods is between 118.9 cm and 119 cm is approximately \(\boxed{0.3970}\).
Step 1 ：The problem is asking for the probability that the average length of a bundle of 40 rods is between 118.9 cm and 119 cm. The lengths of the rods are normally distributed with a mean of 119 cm and a standard deviation of 0.5 cm.
Step 2 ：To solve this, we need to use the properties of the normal distribution. Specifically, we know that the sum (and therefore the average) of a large number of independent and identically distributed random variables follows a normal distribution. This is known as the Central Limit Theorem.
Step 3 ：The mean of this distribution is the same as the mean of the individual variables, and the standard deviation is the standard deviation of the individual variables divided by the square root of the number of variables. So, in this case, the mean of the distribution of the average length of a bundle of 40 rods is 119 cm, and the standard deviation is 0.5 cm divided by the square root of 40.
Step 4 ：We can then use the cumulative distribution function (CDF) of the normal distribution to find the probability that a value is less than a given value. The probability that the average length is between 118.9 cm and 119 cm is the difference between the CDF at 119 cm and the CDF at 118.9 cm.
Step 5 ：Calculating the mean and standard deviation, we get mean = 119 and std_dev = 0.07905694150420949.
Step 6 ：Calculating the CDF at 119 and 118.9, we get cdf_119 = 0.5 and cdf_118_9 = 0.10295160536604708.
Step 7 ：The probability is then calculated as the difference between cdf_119 and cdf_118_9, which gives us probability = 0.3970483946339529.
Step 8 ：Final Answer: The probability that the average length of a bundle of 40 rods is between 118.9 cm and 119 cm is approximately \(\boxed{0.3970}\).
