Answer:
a. To find the probability that it takes at least 6 hours for a randomly selected student to write a history paper, we need to calculate the area under the normal distribution curve to the right of 6 hours.
Using the mean \(\mu = 7.5\) hours and the standard deviation \(\sigma = 1.8\) hours, we can standardize the value of 6 hours using the z-score formula:
\(z = \frac{{x - \mu}}{{\sigma}}\)
\(z = \frac{{6 - 7.5}}{{1.8}}\)
\(z \approx -0.833\)
We can then use a standard normal distribution table or a calculator to find the probability associated with this z-score. The probability of a z-score being less than or equal to -0.833 is approximately 0.2031.
However, we want the probability that it takes at least 6 hours, which is the complement of the probability we just calculated. So, the probability that it takes at least 6 hours is approximately \(1 - 0.2031 = 0.7969\) or 79.69%.
b. To find the probability that the mean time of writing a history paper for a random sample of 16 students is at most 7 hours, we can use the central limit theorem. According to the central limit theorem, the sampling distribution of the sample means approaches a normal distribution with mean equal to the population mean and standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the population mean \(\mu = 7.5\) hours, the population standard deviation \(\sigma = 1.Apologies, it seems that my previous response got cut off. I will continue from where I left off.
b. To find the probability that the mean time of writing a history paper for a random sample of 16 students is at most 7 hours, we can use the central limit theorem. According to the central limit theorem, the sampling distribution of the sample means approaches a normal distribution with a mean equal to the population mean and a standard deviation equal to the population standard deviation divided by the square root of the sample size.
In this case, the population mean \(\mu = 7.5\) hours, the population standard deviation \(\sigma = 1.8\) hours, and the sample size \(n = 16\).
To standardize the value of 7 hours, we can calculate the z-score:
\(z = \frac{{x - \mu}}{{\sigma/\sqrt{n}}}\)
\(z = \frac{{7 - 7.5}}{{1.8/\sqrt{16}}}\)
\(z = \frac{{-0.5}}{{1.8/4}}\)
\(z \approx -0.5556\)
Using a standard normal distribution table or a calculator, we can find the probability associated with this z-score. The probability of a z-score being less than or equal to -0.5556 is approximately 0.2896.
Therefore, the probability that the mean time of writing a history paper for a random sample of 16 students is at most 7 hours is approximately 0.2896 or 28.96%.