## Weights of Pennies

The distribution of weights of US pennies is approximately normal with a mean of 2.5 grams and a standard deviation of 0.03 grams.

(a) What is the probability that a randomly chosen penny weighs less than 2.4 grams?

(b) Describe the sampling distribution of the mean weight of 10 randomly chosen pennies.

(c) What is the probability that the mean weight of 10 pennies is less than 2.4 grams

(d) Sketch the two distributions (population and sampling) on the same scale.

[From OpenIntro Statistics, Second Edition, Problem 4.39]

## Benford’s Law

Assume that the population in a city grows exponentially at rate $$r$$. In other words, the number of people in the city, $$N(t)$$, grows as $$N(t)=C e^{rt}$$, where $$C<10^6$$ is a constant.

1. Determine the time interval $$\Delta t_1$$ during which $$N(t)$$  will be between 1 and 2 million people.

2. For $$k=1,…,9$$, determine the time interval $$\Delta t_k$$ during which $$N(t)$$  will be between k and k+1 million people.

3. Calculate the total time $$T$$ it takes for $$N(t)$$ to grow from 1 to 10 million people.

4. Now pick a time $$\hat t \in [0,T]$$ uniformly at random, and use the above results to derive the following formula (also known as Benford’s law) $$p_k=\mathbb P(N(\hat t) \in [k, k+1] \,million)=\log_{10}(k+1)-\log_{10}(k).$$