you are the operations manager for an airline and you are considering a higher fare level for passengers in…

Explanation:

Step1: Determine the z - value

For a 90% confidence level, the significance level $\alpha=1 - 0.90 = 0.10$. Then $\alpha/2=0.05$. The z - value $z_{\alpha/2}=z_{0.05}\approx1.645$.

Step2: Use the formula for sample size when $p$ is unknown

When nothing is known about the population proportion $p$, we use $p = 0.5$ (this gives the maximum value of $p(1 - p)$). The formula for sample size $n$ in estimating a proportion is $n=\frac{z_{\alpha/2}^{2}\times p(1 - p)}{E^{2}}$, where $E$ is the margin of error. Here, $E = 0.015$ (since 1.5 percentage points = 0.015), and $p = 0.5$, $1-p=0.5$. Substitute the values into the formula: [ \begin{align*} n&=\frac{(1.645)^{2}\times0.5\times(1 - 0.5)}{(0.015)^{2}}\ &=\frac{(1.645)^{2}\times0.25}{(0.015)^{2}}\ &=\frac{2.706025\times0.25}{0.000225}\ &=\frac{0.67650625}{0.000225}\ &\approx 3006.7 \end{align*} ]

Step3: Round up the sample size

Since we cannot have a fraction of a sample, we round up to the nearest integer.

Answer:

3007