**1.**
At a restaurant, the amounts on the checks are i.i.d. with mean 40 dollars and SD 30 dollars. Let $T$ be the total amount on 200 checks and let $M$ be the mean amount on those 200 checks.

**a)** Find $E(T)$ and $SD(T)$.

**b)** If possible, sketch the approximate distribution of $T$ and mark $E(T)$ and $SD(T)$ appropriately on your sketch. If this is not possible, explain why not.

**c)** Find $E(M)$ and $SD(M)$.

**d)** If possible, sketch the approximate distribution of $M$ and mark $E(M)$ and $SD(M)$ appropriately on your sketch. If this is not possible, explain why not.

**2.** Suppose the numbers of M&Ms in the small 1.69-ounce bags of the candy are i.i.d. with mean 55 and SD 2. Let $X$ be the total number of M&Ms in 100 such bags. Find or approximate $P(X > 5525)$.

**3.**
A population consists of 1 million people. Incomes in the population have an average of 70,000 dollars and an SD of 30,000 dollars. A simple random sample of 400 people is taken from the population.

Sketch your best guesses of the shapes of the following histograms and provide your reasoning.

**a)** the histogram of the 1 million incomes in the population

**b)** the histogram of the 400 incomes in the sample

**c)** the probability histogram of the income of a person drawn at random from the population

**d)** the probability histogram of the average of a simple random sample of 400 people drawn from the population

**4.**
**[Continuing the previous exercise.]**

**a)** Fill in the blank with a number: At least 75% of the population has incomes in the range 70,000 dollars plus or minus $\underline{~~~~~~~~~~~~~~~~~~}$ dollars.

**b)** Fill in the blank with a number: With chance about 75%, the average income of the sampled people is in the range 70,000 dollars plus or minus $\underline{~~~~~~~~~~~~~~~~~~}$ dollars.

**5.**
Suppose the weights of sticks of butter are i.i.d. with a mean of 115 grams and an SD of 5 grams. Let $X$ be the total weight of 600 such sticks. Find $x$ such that $P(X > x)$ is approximately 95%.

**6.**
A coin is tossed 100 times. Let $X$ be the number of heads.

**a)** What is the distribution of $X$?

**b)** Sketch the normal curve that approximates the distribution of $X$. Mark the numerical values of $E(X)$ and $SD(X)$ appropriately on the sketch.

**c)** Use the approximation in Part **b** to get a rough numerical value for $P(45 \le X \le 55)$.

**d)** Find the exact numerical value of $P(45 \le X \le 55)$ and compare with the answer to **c**. The approximation is pretty rough.

**e)** The figure below shows the probability histogram of $X$ and the approximating normal curve. Fill in the blanks:

$P(45 \le X \le 55)$ is the area of the bars centered over the integers $\underline{~~~~~~~~~~~~~~}$ through $\underline{~~~~~~~~~~~~~~}$. This is approximately the area under the red curve between the points $\underline{~~~~~~~~~~~~~~}$ and $\underline{~~~~~~~~~~~~~~}$ on the horizontal axis.

**f)** Use Part **e** to get a better normal approximation for $P(45 \le X \le 55)$. This is called the normal approximation *with continuity correction*. You are correcting for using a continuous curve to approximate a discrete histogram.