Question

Will there be a time when hiring decisions will be done using a computer algorithm, and without any human involvement? A Pew Research survey found that \(67 \%\) of the adults surveyed were worried about the development of algorithms that can evaluate and hire job candidates. \({ }^{12}\) To check this claim, a random sample of \(n=100\) U.S. adults was selected, with 58 saying they were worried about this issue. a. Construct a \(98 \%\) confidence interval for the true proportion of adults that are worried about being "hired by a robot". b. Does the confidence interval constructed in part a confirm the claim of the Pew Research survey? Why or why not?

Short answer

Based on a sample of 100 U.S. adults, we found that 58% are worried about being "hired by a robot." We then calculated a 98% confidence interval for the true proportion of adults concerned about this issue, which was found to be between 45.22% and 70.78%. Because the Pew Research survey's claim of 67% lies within this confidence interval, our results confirm and do not contradict their claim.

Step-by-step solution

Identify the necessary information

We are given: - Sample size: \(n=100\) - Number of worried people in the sample: 58 - Confidence level: 98% First, we need to calculate the sample proportion \(\widehat{p}\) and the standard error of the proportion.

Calculate the sample proportion

To find the sample proportion \(\widehat{p}\), divide the number of worried people (58) by the total sample size (100): \(\widehat{p}=\frac{58}{100}=0.58\) This means that 58% of people in our sample are worried about being "hired by a robot".

Calculate the standard error

Next, we'll calculate the standard error (SE) of the proportion using the formula: \(SE=\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\) Plug in the values: \(SE=\sqrt{\frac{(0.58)(1-0.58)}{100}}=\sqrt{\frac{(0.58)(0.42)}{100}}\approx 0.0496\)

Find the critical value

To find the critical value, we'll use the Z-table. Since we want a 98% confidence interval, we'll look for the Z-value that corresponds to the middle 98% of the distribution, which leaves 1% on each tail. So, our Z-value will correspond to a 99% cumulative probability: \(Z_{\alpha/2}=2.576\)

Calculate the margin of error

Now, we'll calculate the margin of error (ME) using the formula: \(ME=Z_{\alpha/2}\times SE\) Plug in the values: \(ME=2.576\times 0.0496\approx 0.1278\)

Construct the confidence interval

Finally, we'll construct the 98% confidence interval for the true proportion of adults that are worried about being "hired by a robot" using the formula: CI: \((\widehat{p}-ME,\widehat{p}+ME)\) Plug in the values: CI: \((0.58-0.1278,0.58+0.1278)=(0.4522, 0.7078)\) Therefore, we are 98% confident that the true proportion of adults that are worried about being "hired by a robot" lies between 45.22% and 70.78%.

Answer part b

The Pew Research survey claim was that 67% of adults surveyed were worried about the development of algorithms that can evaluate and hire job candidates. In part a, we calculated the 98% confidence interval for our sample to be (45.22%, 70.78%). Since 67% lies within this interval, our results do not contradict the Pew Research survey's claim. Therefore, the confidence interval constructed in part a does confirm the claim of the Pew Research survey.

Key concepts

Sample Proportion

The sample proportion is a crucial concept when analyzing data from surveys or experiments. It represents the ratio of individuals in a sample with a particular characteristic to the total number of individuals in the sample. In the context of our exercise, the sample proportion, denoted as \(\widehat{p}\), was calculated to find the percentage of people in a sample of 100 U.S. adults who were concerned about being 'hired by a robot'. To find \(\widehat{p}\), we took the number of individuals worried about the issue (58) and divided by the total number of individuals surveyed (100), resulting in a sample proportion of 0.58, or 58%. This figure is an estimate of the true population proportion and serves as a starting point for further statistical analysis.

Understanding the sample proportion is essential as it helps determine the likelihood of an event or characteristic in a broader population, based on the sampled data. It's the foundation from which the confidence interval is constructed, which aims to capture the true population proportion within a specified degree of certainty.

Standard Error

The standard error (SE) measures the precision of the sample proportion as an estimate of the population proportion. It is essentially the standard deviation of the sampling distribution of the sample proportion. Calculating the SE helps us understand the variability we might expect in sample proportions if we were to take multiple samples from the same population under the same conditions.

To calculate the SE in our exercise, we used the formula \(SE=\sqrt{\frac{\widehat{p}(1-\widehat{p})}{n}}\), incorporating the sample proportion \(\widehat{p}\) and the sample size \(n\). This calculation yielded an SE of approximately 0.0496, giving us an estimate of the extent to which the sample proportion \(0.58\) might differ from the actual population proportion due to sampling variability.

Margin of Error

The margin of error (ME) plays a pivotal role in understanding the range within which the true population proportion is likely to fall. It combines the standard error with a Z-value (discussed next) to provide the bounds around our sample proportion estimate that we might expect to contain the true population proportion a certain percentage of the time.

In our exercise, we multiplied the Z-value associated with a 98% confidence level by the standard error to find the ME: \(ME=Z_{\alpha/2}\times SE\). This calculation provided a ME of approximately 0.1278, which means we can be pretty confident that if we repeatedly sampled from the population, the true population proportion of concerned individuals would fall within 12.78 percentage points of our sample proportion most of the time.

Z-value

A Z-value, also known as a Z-score, is a measure that describes a value's relationship to the mean of a group of values, expressed in terms of standard deviations from the mean. When constructing confidence intervals for a sample proportion, the Z-value is used to determine how far out from the sample proportion we extend our interval, based on the desired level of confidence.

For our 98% confidence interval, we looked for the Z-value that left 1% in each tail of the normal distribution, corresponding to a cumulative probability of 99%. This resulted in a Z-value of 2.576. This value indicates that our margin of error extends 2.576 standard deviations from the sample proportion, capturing 98% of the possible sample proportions if the sampling were repeated many times.

Statistical Significance

Statistical significance is a term used to determine whether the results of a study or experiment are unlikely to have occurred by chance and can support a hypothesis. It's often associated with a p-value, which is the probability of observing a result at least as extreme as the one obtained given that the null hypothesis is true.

In the context of our confidence interval, statistical significance isn't directly assessed through a p-value. However, the confidence level associated with the interval (in this case, 98%) gives us an idea of how confident we are that the interval includes the true population proportion. The fact that the Pew Research claim of 67% concerned adults falls within our calculated interval suggests that the claim is consistent with the data obtained from our sample, implying that there is a significant likelihood that the true proportion of concerned individuals could be around 67%.