If a random variable X equals 2 with probablity 0.386, if i make a repeated independent observations of the random variable X, in approximately what percentage of those observe the value 2? and roughly how many times would i expect to observe the value 2 in 50 observations? and roughly how many times would i expect to observe 2 in 500 observations?

If X; and independent, identically distributed (iid) Gaussian random variables with mean, µ, and variance, o², for i € {1, ..., n}, what is the distribution of the sample mean? How is the sample variance related to a common distribution? Sketch the distributions from problem 3 and label key points.

5. If z is a standard normal random variable, find the value of c:P(z < c)= 0.6950                    c = _____

1.The U.S. Geological Survey compiled historical data about Old Faithful Geyser (Yellowstone National Park) from 1870 to 1987. Let x1 be a random variable that represents the time interval (in minutes) between Old Faithful eruptions for the years 1948 to 1952. Based on 8880 observations, the sample mean interval was x1 = 63.0 minutes. Let x2 be a random variable that represents the time interval in minutes between Old Faithful eruptions for the years 1983 to 1987. Based on 25,691 observations, the sample mean time interval was x2 = 70.2 minutes. Historical data suggest that ?1 = 8.70 minutes and ?2 = 12.48 minutes. Let ?1 be the population mean of x1 and let ?2 be the population mean of x2. (a) Compute a 95% confidence interval for ?1 – ?2. (Use 2 decimal places.) lower limit   upper limit   (b) Comment on the meaning of the confidence interval in the context of this problem. Does the interval consist of positive numbers only? negative numbers only? a mix of positive and…