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When we talk about population proportions, we're referring to the fraction of the total population that has a particular characteristic. In the context of this exercise, this characteristic is playing the lottery often. The samples taken from the population help us estimate the unknown true proportion for each group (men and women).

Here's how you calculate them: For men, with 160 out of 500 playing often, the proportion is 160 divided by 500, resulting in 0.32. For women, where 66 out of 300 play often, divide 66 by 300 to get 0.22. These proportions tell us that 32% of our sample of men and 22% of the sample of women play the lottery frequently. These sample proportions help us form a point estimate, which is simply

• Proportion of men: \(P_m = 0.32\)
• Proportion of women: \(P_w = 0.22\)

A confidence interval gives us a range of values that are likely to contain the true difference between two population proportions. This helps us understand the precision of our estimate and how much it might vary from the actual population value.

For this problem, we used a 99% confidence interval to estimate the difference in lottery-playing proportions between men and women. The confidence level tells us how sure we can be that this interval includes the real difference. A 99% confidence level means we can be very confident, but it's also wider than, say, a 95% confidence interval. To calculate:

• Use the formula \[(P_m - P_w) \pm Z*\sqrt{\left(\frac{P_m(1-P_m)}{n_m}\right) + \left(\frac{P_w(1-P_w)}{n_w}\right)}\]
• The Z-value for 99% confidence is 2.58.
• Substitute the numbers: \[0.1 \pm 2.58*\sqrt{\frac{0.32*0.68}{500} + \frac{0.22*0.78}{300}}\]

This calculation will provide a range of values we believe the true difference falls into, considering our high level of confidence.

The significance level, often denoted by \( \alpha \), is a threshold used in statistical tests to determine whether to reject the null hypothesis. In this exercise, a 1% significance level was chosen. This means that we tolerate only a 1% chance of concluding that there's a difference when there actually isn't.

Here's the deal:

• The null hypothesis (\(H_0\)) is that the proportion of men playing the lottery often is the same as for women.
• The alternative hypothesis (\(H_1\)) is that these proportions are not equal.
• We set a very strict \( \alpha \) at 0.01 because we want to be very certain before claiming a significant difference exists.

If our test results (p-value) fall below this significance level, we reject the null hypothesis, thereby supporting the idea that there's a difference.

The p-value in hypothesis testing measures how likely we are to see the observed data, or something more extreme, assuming the null hypothesis is true. It's a critical value you compare against the significance level to make decisions.

To compute the p-value, we follow these steps:

• First, calculate the test statistic using the formula \[\frac{P_m - P_w}{\sqrt{P(1-P)\left(\frac{1}{n_m} + \frac{1}{n_w}\right)}}\]
• Here, \(P\) is the pooled sample proportion, which gives an average based on all individuals surveyed.
• Once you have this statistic, compare it to a standard normal distribution to get the p-value.

If this p-value is less than 0.01 in this exercise, it indicates a statistically significant difference between men's and women's lottery-playing habits, allowing us to reject the null hypothesis confidently.