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In many statistical problems, we are interested in understanding how a sample proportion behaves. A sample proportion is simply the proportion observed in a sample rather than in the whole population. For instance, if we say 70% of drivers exceed the speed limit, this 70% is our population proportion. However, when we select 80 drivers to check their speeds, the speeders among these 80 drivers give us a sample proportion. The sample proportion is denoted by \( \hat{p} \). In our exercise, the state police are trying to see how this sample proportion (of drivers speeding) will behave statistically. To do this, they look at the distribution of sample proportions which can be approximated to a normal distribution if certain conditions are met.

Normal distribution is a statistical function that describes how values are dispersed in a dataset. It is represented by a bell-shaped curve, known as a Gaussian distribution. For our scenario with the state police, if we know the underlying data follows a normal distribution, or can approximate such a curve, this helps us make predictions about data behavior.In the context of our exercise, the goal is to figure out the normal distribution of the sample proportion, \( \hat{p} \). This distribution will be centered around the population proportion \( p = 0.7 \), with a spread determined by a formula for standard deviation (which we'll discuss more in the next sections). The benefit of knowing this is that it makes it easier to calculate probabilities and understand the variability of the sample proportion.

The 68-95-99.7 rule is a handy shorthand that helps us quickly understand the distribution of data in a normal distribution. This rule simply states:- 68% of all outcomes will fall within one standard deviation of the mean.- 95% of values will fall within two standard deviations.- Nearly all, about 99.7%, will fall within three standard deviations.For the police exercise, if they want to visualize the distribution of the sample proportion of speeding drivers, this rule gives them boundaries to look for. Centered at \( p = 0.7 \), and using the calculated standard deviation \( \sigma_{\hat{p}} = 0.0516 \), they can use these guidelines to set intervals like \( 0.7 \pm 0.0516 \), \( 0.7 \pm 2 \times 0.0516 \), and \( 0.7 \pm 3 \times 0.0516 \). This helps in understanding variations and making decisions confidently.

To apply the central limit theorem (CLT) and assume the sampling distribution of \( \hat{p} \) is normal, we need certain conditions to be met. Before using a normal approximation for a sample proportion, two key checks are necessary:- The product \( np \geq 10 \), ensuring a sufficiently large sample size of 'successes'.- The product \( n(1-p) \geq 10 \), confirming there's a sufficiently large sample size of 'failures'.In our scenario, \( np = 56 \) and \( n(1-p) = 24 \) satisfy these conditions (as both exceed 10). This means the sample size is adequate to safely assume that the sampling distribution of the sample proportion (\( \hat{p} \)) follows a normal distribution. When these criteria are met, it justifies the use of a normal model to make statistical inferences on, say, how many drivers in future samples might speed.