Calculate T Tests And Confidence Intervals For The Following

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Calculate T Tests And Confidence Intervals For The Following Sets Of I Calculate t tests and confidence intervals for the following sets of information, using the confidence level indicated. Then comment on statistical significance, say whether we reject or fail to reject the null, and tell me how you know. 1. $\hat{\beta} = 2.94497$, standard error $= 4.863653$, $n=580$, independent variables $= 2$, use 95% confidence level. 2. $\hat{\beta} = -19.3$, standard error $= 0.6807$, $n=5$, independent variables $= 1$, use 95% confidence level. 3. Murder rate $= 0.348 + 0.165(\text{EXEC}) + 1.26(\text{UNEMPL})$ with standard errors $(2.69), (1.94), (0.437)$, $n=153$, use 99% confidence interval for both variables. Suppose you work for an insurance company, and you sell a \$120,000 fire insurance policy at an annual premium of \$1250. Experience has shown that: - The probability of total loss (due to fire) on a house in that area and of the size of your customer's house is 0.001 (full payout of \$120,000). - The probability of 50% damage (due to fire) is 0.003 (payout of \$60,000). Ignore other partial losses for simplicity. (a) Write down the probability distribution of $X$, the insurance company's annual gain from such a policy. (b) What is the mean (expected) annual gain for a policy of this type? (c) What should be the annual premium (instead of \$1250) if the company wants to increase the expected gain from a policy of this type by 10%?

Dr. Jack HW Helper · Accepted Answer

The analysis of t tests, confidence intervals, and their implications for statistical significance forms a fundamental aspect of inferential statistics. When evaluating regression coefficients such as $\hat{\beta}$, t tests determine whether these coefficients significantly differ from zero, implying a meaningful relationship between variables. Confidence intervals provide a range within which the true parameter value is likely to fall, with a specified degree of certainty. Together, these tools aid researchers and analysts in making informed decisions based on empirical data. Part 1: T Test and Confidence Interval for $\hat{\beta} = 2.94497$ Given $\hat{\beta} = 2.94497$, standard error $= 4.863653$, and a sample size $n=580$, with two independent variables, the t statistic is calculated as: $$ t = \frac{\hat{\beta}}{se} = \frac{2.94497}{4.863653} \approx 0.605 $$ This t value is compared against the critical t value for a 95% confidence level with degrees of freedom $df = n - p - 1 = 580 - 2 - 1 = 577$. From t-distribution tables, the critical value is approximately 1.96 for a two-tailed test. Since $|t| = 0.605 \[ \hat{\beta} \pm t_{critical} \times se = 2.94497 \pm 1.96 \times 4.863653 \] \[ = 2.94497 \pm 9.5357 \] \[ = (-6.5907, 12.4807) \] This interval contains zero, further confirming that \(\beta$ is not statistically significant at the 95% level. Part 2: T Test and Confidence Interval for $\hat{\beta} = -19.3$ With $\hat{\beta} = -19.3$, $se = 0.6807$, $n=5$, and one independent variable, the t-statistic is: $$ t = \frac{-19.3}{0.6807} \approx -28.36 $$ Degrees of freedom $df = n - p - 1= 5 - 1 - 1=3$. The critical t value at 95% confidence with 3 df is approximately 3.182. Since $|t|=28.36 > 3.182$, we reject the null hypothesis, indicating $\beta$ is statistically significant. The 95% confidence interval is: $$ -19.3 \pm 3.182 \times 0.6807 $$ $$ = -19.3 \pm 2.165 $$ $$ = (-21.465, -17.135) $$ Because zero is not in this in

Calculate T Tests And Confidence Intervals For The Following Sets Of I

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