## Given The Simple Regression Model Y=b0x+b1x And The Regression Results

The given text snippets describe various types of calculators that can be used to calculate z-scores and probabilities for a normal distribution. The calculators can also convert between z-score and probability, find the probability between two z-scores, determine the standardized z-score for a data point, and find the area under a standardized normal distribution curve. Step-by-step explanations and input values for mean, standard deviation, and data point are provided. There is also an additional calculator for finding critical values of t, f, and chi-square.

Based on the information provided, we can first calculate the z-score and then determine the p-value for each scenario to test the null hypothesis that the slope coefficient is 0 versus the alternative hypothesis of greater than zero.

a. Sample size, n = 36 b1 = 7 sb1 = 1.6 Given a probability of Type 1 error, α = 0.01

We can calculate the z-score: z = (b1 - 0) / sb1 = (7 - 0) / 1.6 = 4.375

Looking up the z-score 4.375 in the z-table, the corresponding p-value is essentially 0. Therefore, we reject the null hypothesis. There is sufficient evidence that the slope coefficient is greater than zero.

For the two-sided 95% confidence interval, we find the critical z-value at α/2 = 0.025, which is approximately 1.96. So, the interval is (7 - 1.96 * 1.6, 7 + 1.96 * 1.6) = (3.92, 10.08).

For the two-sided 99% confidence interval, we find the critical z-value at α/2 = 0.005, which is approximately 2.58. So, the interval is (7 - 2.58 * 1.6, 7 + 2.58 * 1.6) = (2.46, 11.54).

b. Sample size, n = 47 b1 = 7.2 sb1 = 1.6 Given a probability of Type 1 error, α = 0.01

We can calculate the z-score: z = (b1 - 0) / sb1 = (7.2 - 0) / 1.6 = 4.5

Looking up the z-score 4.5 in the z-table, the corresponding p-value is essentially 0. Therefore, we reject the null hypothesis. There is sufficient evidence that the slope coefficient is greater than zero.

For the two-sided 95% confidence interval, we find the critical z-value at α/2 = 0.025, which is approximately 1.96. So, the interval is (7.2 - 1.96 * 1.6, 7.2 + 1.96 * 1.6) = (3.04, 11.36).

For the two-sided 99% confidence interval, we find the critical z-value at α/2 = 0.005, which is approximately 2.58. So, the interval is (7.2 - 2.58 * 1.6, 7.2 + 2.58 * 1.6) = (1.84, 12.56).

c. Sample size, n = 36 b1 = 4.3 sb1 = 1.15 Given a probability of Type 1 error, α = 0.01

We can calculate the z-score: z = (b1 - 0) / sb1 = (4.3 - 0) / 1.15 = 3.7391

Looking up the z-score 3.7391 in the z-table, the corresponding p-value is approximately 0.9998. Therefore, we do not reject the null hypothesis. There is insufficient evidence that the slope coefficient is greater than zero.

For the two-sided 95% confidence interval, we find the critical z-value at α/2 = 0.025, which is approximately 1.96. So, the interval is (4.3 - 1.96 * 1.15, 4.3 + 1.96 * 1.15) = (2.14, 6.46).

For the two-sided 99% confidence interval, we find the critical z-value at α/2 = 0.005, which is approximately 2.58. So, the interval is (4.3 - 2.58 * 1.15, 4.3 + 2.58 * 1.15) = (1.09, 7.51).

d. Sample size, n = 34 b1 = 9.4 sb1 = 1.1 Given a probability of Type 1 error, α = 0.01

We can calculate the z-score: z = (b1 - 0) / sb1 = (9.4 - 0) / 1.1 = 8.5455

Looking up the z-score 8.5455 in the z-table, the corresponding p-value is essentially 0.

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