How do I tell if this is a mean or a proportion

1. Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. The insurance company knows that, last year, the life expectancy of its
policyholders was 77 years. They want to know if their clients this year have a longer life expectancy, on average, so the company randomly samples some of the recently paid policies to see if the mean life expectancy of policyholders has increased. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before.

Ages:
86 75 83 84 81 77 78 79 79 81 76 85 70 76 79 81 73 74 72 83

Does this sample indicate that the insurance company should change its premiums because life expectancy has increased? Test an appropriate hypothesis and state your conclusion. Find and interpret a 95% confidence interval.

And how do I even start this question

It is a mean, or average. The company is using random samples to get the mean life expectancy of its policyholders.

To determine whether the insurance company should change its premiums because life expectancy has increased, you need to perform a hypothesis test and calculate a confidence interval. Here's how you can approach this question:

1. Hypothesis Testing:
- State the null hypothesis (H₀) and the alternative hypothesis (H₁).
- The null hypothesis (H₀) is typically a statement of no effect or no difference.
- The alternative hypothesis (H₁) is the opposite of the null hypothesis and represents the research question or what you want to prove.
- In this case, the null hypothesis (H₀) would be that the mean life expectancy of policyholders this year is the same as last year, and the alternative hypothesis (H₁) would be that the mean life expectancy has increased.

2. Test Statistic:
- To test the hypothesis, you can calculate the test statistic, which compares the sample mean to the population mean from the previous year.
- In this case, you would calculate the sample mean of the ages and compare it to 77 (last year's life expectancy).

3. Critical Value or P-value:
- Depending on the question, you'll either use critical values or calculate p-values.
- For a significance level of 0.05 (95% confidence level), you'll compare the test statistic against the critical value from the appropriate distribution (e.g., t-distribution for small sample sizes).

4. Conclusion:
- Based on the test statistic and the critical value or p-value, you'll decide whether to reject or fail to reject the null hypothesis.
- If the test statistic falls outside the critical region (i.e., exceeds the critical value) or the p-value is less than the significance level, you'll reject the null hypothesis and conclude that there is evidence of an increase in life expectancy, providing support to change the insurance company's premiums.

5. Confidence Interval:
- To calculate a 95% confidence interval, you can use the sample mean, sample standard deviation, and the appropriate critical value from the t-distribution.
- The confidence interval will give you a range of values within which you can be 95% confident that the true population mean lies.

By following these steps, you can perform the hypothesis test and calculate the confidence interval to answer the question about whether the insurance company should change its premiums. Remember to check for any assumptions or conditions that might need to be met for hypothesis testing.