Question

1. A random number generator claims to randomly choose real numbers between 0 and 3000. (a) If this is true, what kind of distribution would the randomly chosen numbers have? (b) What would be the mean of this distribution? (c) What would be the standard deviation of this distribution? (d) Say we take a sample of 65 generated numbers and obtain a sample mean of 1552. What do we know about the sampling distribution of the sample mean and how do we know this? (e) Give a 99% confidence interval about the population mean. (f) In a sentence or two, state what the 99% confidence interval from(e) tells us. (g) Suppose that we do not know the population standard deviation and, instead,the sample standard deviation is s = 822. What is the confidence interval about the population mean using the sample standard deviation? Compare this to your answer in part (e).

Answer 1

Solution

(a) Continuous Uniform Distribution. Parameters are

(b) The mean is

(c) The variance is

Hence, the standard deviation is

(d) Regardless of the shape of the parent population, the sampling distribution of the mean approaches a normal distribution as sample size increases. For N = 65, normal approximation can be applied.

Hence, The sample mean follows a normal distribution with mean equal to µ, and standard deviation (standard error) equal to

For given values,

(e) The standard error of mean is

The Z-score for 99% confidence interval is ± 2.576. Hence, the interval is obtained as

(f) If the sample means are computed for a sample size of 65, we expect that about 99% times, the mean shall lie between 1275.31155 and 1828.68845

(g) The population mean being unknown, the confidence interval is obtained using a t-distribution of the mean with n-1 = 64 degrees of freedom

The critical t-statistic is

Hence the confidence interval is

This is a slightly narrow interval, because observed sample standaard deviation is slightly less than the theoretical value