1. (Based on the data from textbook problem 1, page 54) The data in Table 3.3 are a subset of the data obtained by Kaneto, Kosaka, and Nakao (1967). The experiment investigated the effect of vagal nerve stimulation on insulin secretion. The subjects were mongrel dogs with varying body weights. Table 3.3 gives the amount of immunoreactive insulin in pancreatic venous plasma just before stimulation of the left vagus nerve (X) and the amount measured 5 min after stimulation (Y) for seven dogs. Test the hypothesis of no effect against the alternative that stimulation of the vagus nerve increases the blood level of immunoreactive insulin. Table 3.3 Blood Levels of Immunoreac- tive Insulin (U/ml) Dog i 1 2 3 4 5 6 7 X₁ 350 200 240 290 90 370 240 Y₁ 480 130 250 310 280 1450 280 Source: A. Kaneto, K. Kosaka, and K. Nakao (1967).

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(a) Write down the appropriate null and alternative hypotheses.

(b) Provide a table that contains the values of \( Z_i, |Z_i|, R_i, \psi_i, \) and \( R_i \psi_i \) for \( i = 1, 2, \ldots, n \), for this data. Then calculate the Wilcoxon signed rank test statistic \( T^+ \). Show your work. (e.g., See Table 3.1 for Example 3.1 in our Note-3.)

(c) Find the integer \( t_{\alpha} = t_{0.0391} \) such that \( P(T^+ \geq t_{\alpha}) = 0.0391 \) (rounded the probability to four decimal places). Use “psignrank” in R.

(d) Test the hypotheses using the Wilcoxon signed rank test, keeping the significance level at no more than 0.05. Find the p-value and give your conclusion in the context of the problem.

(e) Test the hypotheses using the large-sample approximation. Give your test statistic value and approximate p-value. State your conclusion in the context of the problem using \( \alpha = 0.05 \).
Transcribed Image Text:(a) Write down the appropriate null and alternative hypotheses. (b) Provide a table that contains the values of \( Z_i, |Z_i|, R_i, \psi_i, \) and \( R_i \psi_i \) for \( i = 1, 2, \ldots, n \), for this data. Then calculate the Wilcoxon signed rank test statistic \( T^+ \). Show your work. (e.g., See Table 3.1 for Example 3.1 in our Note-3.) (c) Find the integer \( t_{\alpha} = t_{0.0391} \) such that \( P(T^+ \geq t_{\alpha}) = 0.0391 \) (rounded the probability to four decimal places). Use “psignrank” in R. (d) Test the hypotheses using the Wilcoxon signed rank test, keeping the significance level at no more than 0.05. Find the p-value and give your conclusion in the context of the problem. (e) Test the hypotheses using the large-sample approximation. Give your test statistic value and approximate p-value. State your conclusion in the context of the problem using \( \alpha = 0.05 \).
### Vagal Nerve Stimulation and Insulin Secretion in Dogs

The following section explores an experiment conducted by Kaneto, Kosaka, and Nakao (1967) to investigate the effect of vagal nerve stimulation on insulin secretion. The subjects were mongrel dogs with varying body weights.

#### Objectives
- To observe changes in immunoreactive insulin levels in pancreatic venous plasma before and after stimulation of the left vagus nerve.
- To test the hypothesis that stimulation of the vagus nerve increases the blood level of immunoreactive insulin.

#### Data Summary
The table below presents the blood levels of immunoreactive insulin measured in micro-units per milliliter (μU/ml) for seven dogs. For each dog, \( X_i \) denotes the insulin level before stimulation, and \( Y_i \) denotes the level measured five minutes after stimulation.

| Dog \( i \) | \( X_i \) (Before Stimulation) | \( Y_i \) (After Stimulation) |
|-------------|-------------------------------|------------------------------|
| 1           | 350                           | 480                          |
| 2           | 200                           | 130                          |
| 3           | 240                           | 250                          |
| 4           | 290                           | 310                          |
| 5           | 90                            | 280                          |
| 6           | 370                           | 1450                         |
| 7           | 240                           | 280                          |

**Source:** A. Kaneto, K. Kosaka, and K. Nakao (1967).

#### Analysis
The hypothesis to be tested is whether there is no effect of vagus nerve stimulation compared to an increase in insulin levels following stimulation. The data illustrates varying levels of insulin in response to the nerve stimulation, with some dogs exhibiting significant increases. Further statistical analysis is required to draw concrete conclusions.

This dataset provides crucial insights into understanding the physiological effects of neural stimulation on insulin secretion, potentially informing medical interventions and therapies.
Transcribed Image Text:### Vagal Nerve Stimulation and Insulin Secretion in Dogs The following section explores an experiment conducted by Kaneto, Kosaka, and Nakao (1967) to investigate the effect of vagal nerve stimulation on insulin secretion. The subjects were mongrel dogs with varying body weights. #### Objectives - To observe changes in immunoreactive insulin levels in pancreatic venous plasma before and after stimulation of the left vagus nerve. - To test the hypothesis that stimulation of the vagus nerve increases the blood level of immunoreactive insulin. #### Data Summary The table below presents the blood levels of immunoreactive insulin measured in micro-units per milliliter (μU/ml) for seven dogs. For each dog, \( X_i \) denotes the insulin level before stimulation, and \( Y_i \) denotes the level measured five minutes after stimulation. | Dog \( i \) | \( X_i \) (Before Stimulation) | \( Y_i \) (After Stimulation) | |-------------|-------------------------------|------------------------------| | 1 | 350 | 480 | | 2 | 200 | 130 | | 3 | 240 | 250 | | 4 | 290 | 310 | | 5 | 90 | 280 | | 6 | 370 | 1450 | | 7 | 240 | 280 | **Source:** A. Kaneto, K. Kosaka, and K. Nakao (1967). #### Analysis The hypothesis to be tested is whether there is no effect of vagus nerve stimulation compared to an increase in insulin levels following stimulation. The data illustrates varying levels of insulin in response to the nerve stimulation, with some dogs exhibiting significant increases. Further statistical analysis is required to draw concrete conclusions. This dataset provides crucial insights into understanding the physiological effects of neural stimulation on insulin secretion, potentially informing medical interventions and therapies.
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