EXAMPLE 8.2

Inference for robotics and jobs. The sample survey in Example 8.1 found that 910 of a sample of 1896 experts reported that they think net jobs will decrease by 2025 because of robots and related technology developments. Thus, the sample size is n = 1896 and the count is X = 910. The sample proportion is

$$\widehat{p}=\frac{X}{n}=\frac{910}{1896}=0.47996$$

The standard error is

$${\text{SE}}_{\widehat{p}}=\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}=\sqrt{\frac{0.47996(1-0.47996)}{1896}}=0.011474$$

The z critical value for 95% confidence is z* = 1.96, so the margin of error is

$$m=1.96{\text{SE}}_{\widehat{p}}=\left(1.96\right)\left(0.011474\right)=0.022489$$

The confidence interval is

$$\widehat{p}\pm m=0.480\pm 0.022$$

We are 95% confident that between 45.8% and 50.2% of CEOs would report that they think net jobs will decrease by 2025 because of robots and related technology developments.