Consider two populations for which μ1 = 35, σ1 = 3, μ2 = 20, and σ2 = 4. Suppose that two independent random samples of sizes n1 = 47 and n2 = 54 are selected. Describe the approximate sampling distribution of x1 − x2 (center, spread, and shape). What is the shape of the distribution? The distribution would be non-normal. The distribution is approximately normal. The shape cannot be determined. What is the mean of the distribution? What is the standard deviation of the distribution?

The distribution is approximately normal.

The mean and standard deviation are 15 and 0.98 respectively.

Step-by-step explanation:

The population of the random variables X₁ and X₂ are distributed as follows:

Two independent random samples of sizes,

are selected form the two populations.

According to the Central Limit Theorem if we have an unknown population with mean μ and standard deviation σ and appropriately huge random samples (n > 30) are selected from the population with replacement, then the distribution of the sample mean will be approximately normally distributed.  

Then, the mean of the sample means is given by,

And the standard deviation of the sample means is given by,

Both the sample selected from the two populations are quite large, i.e.

So, according to the central limit theorem the sampling distribution of sample means can be approximated by the Normal distribution.

Then, the distribution of is:

And the distribution of is:

If two random variables are normally distributed then their linear function is also normally distributed.

So, the distribution of is Normal and the shape of the distribution is bell-shaped.

The mean of the distribution of is:

The standard deviation of the distribution of is:

*X₁ and X₂ are independent.

Thus, the mean and standard deviation are 15 and 0.98 respectively.

f(x) has value 150 at x= 0

and 0 at x= 3

so d(f(x))/dt = slope = -150/3 = -50

345

step-by-step explanation: