The time between successive customers coming to the market X is assumed to have Exponential distribution with parameter I.

\(\displaystyle{X}\approx{E}{x}{p}{\left({I}\right)}\)

a)

The mean of exponential distribution with parameter I is I.

As the sample mean is unbiased estimate of population mean we say the following

\(\displaystyle\frac{1}{\hat{{I}}}=\frac{{\sum{x}_{{i}}}}{{n}}\)

This is the estimate for the parameter of the distribution.

b)

The data given 12 randomly selected times between successive customers as shown below

\(\displaystyle{1.8},{1.2},{0.8},{1.4},{1.2},{0.9},{0.6},{1.2},{1.2},{0.8},{1.5},{0.6}\)

The parameter is estimated as shown below

\(\displaystyle\frac{1}{\hat{{I}}}=\frac{{\sum{x}_{{i}}}}{{n}}\)

\(\displaystyle\frac{1}{\hat{{I}}}=\frac{{{1.8},{1.2},{0.8},{1.4},{1.2},{0.9},{0.6},{1.2},{1.2},{0.8},{1.5},{0.6}}}{{12}}\)

\(\displaystyle\hat{{I}}=\frac{10}{{11}}={0.9091}\)

So the exponential distribution is shown below

\(\displaystyle{P}{\left({X}={x}\right)}={I}{e}^{{{I}{x}}}{x}\ge{0}\)

c)

Given

The margin of error \(\displaystyle{E}={0.3}\)

Level of significance \(\displaystyle={4}\%\)

If X is exponential distribution with parameter I then X is approximately normal distribution with mean and variance \(\displaystyle\frac{1}{{I}}{\quad\text{and}\quad}{\left(\frac{1}{{I}}\right)}^{2}\) respectively.

We can approximate the distribution is normal with mean and variance as \(\displaystyle\frac{10}{{11}},{\left(\frac{10}{{11}}\right)}^{2}\) respectively.

Thus the Z critical score for \(4\%\) level of significance or \(96\%\) confidence interval is

\(\displaystyle{Z}_{{{1}-{0.04}\text{/}{2}}}={Z}_{{0.98}}={2.05}\)

The sample size required is calculated as shown below

\(\displaystyle{E}=\frac{{{Z}\sigma}}{\sqrt{{n}}}\)

\(\displaystyle{n}={\left(\frac{{{Z}\sigma}}{{E}}\right)}^{2}\)

\(\displaystyle{n}={\left(\frac{{\sqrt[{{2.05}}]{{\frac{10}{{11}}}}}}{{0.3}}\right)}^{2}\)

\(\displaystyle{n}={42.45}\)

Thus we can say the minimum sample size required is 43. (rounding to next integer).