A survey found that in 
2
0
0
8
,
 ﻿doctors in residency programs at U
.
S
.
 ﻿hospitals worked an average of 
5
9
.
3
 ﻿hours per week. A new residency program guideline went into effect in 
2
0
0
9
,
 ﻿with the aim of reducing the number of hours residents work per week. In 
2
0
0
9
,
 ﻿a small survey of 
2
0
 ﻿residents found the following weekly hours worked:
5
3
,
 
5
7
,
 
5
2
,
 
6
5
,
 
7
0
,
 
5
1
,
 
5
5
,
 
6
1
,
 
4
9
,
 
6
1
,
5
4
,
 
6
4
,
 
4
4
,
 
5
7
,
 
5
5
,
 
6
0
,
 
5
3
,
 
6
1
,
 
5
1
,
 
5
6
You would like to show that that the new guidelines have reduced the number of hours worked per week. Perform a test using alpha
=
 
0
.
0
5
,
 ﻿and answer the following:
a
.
 ﻿State the Null and Alternative hypotheses
b
.
 ﻿Does the sample provide enough evidence to suggest that the new guidelines have reduced the number of hours worked per week?
c
.
 ﻿For the test in 
(
b
)
,
 ﻿what is the p
-
value?

To determine whether the new guidelines have reduced the average number of hours worked per week, we will perform a **one-sample t-test** since the sample size is small (\(n = 20\)) and the population standard deviation is unknown. Let’s proceed step by step.

---

### a. Null and Alternative Hypotheses
1. **Null Hypothesis (\(H_0\))**: The average weekly hours worked under the new guidelines is the same as or more than before:
   \[
   H_0: \mu \geq 59.3
   \]
2. **Alternative Hypothesis (\(H_a\))**: The average weekly hours worked under the new guidelines has been reduced:
   \[
   H_a: \mu < 59.3
   \]

This is a one-tailed test since we are testing for a reduction.

---

### b. Does the sample provide enough evidence to suggest that the new guidelines have reduced the number of hours worked per week?

1. **Given Data:**
   - Sample size (\(n\)) = 20
   - Sample data: \(53, 57, 52, 65, 70, 51, 55, 61, 49, 61, 54, 64, 44, 57, 55, 60, 53, 61, 51, 56\)
   - Population mean (\(\mu_0\)) = 59.3
   - Significance level (\(\alpha\)) = 0.05

2. **Calculate the sample mean (\(\bar{x}\)):**
   \[
   \bar{x} = \frac{\text{Sum of all sample values}}{n}
   \]

3. **Calculate the sample standard deviation (\(s\)):**
   \[
   s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}
   \]

4. **Compute the t-statistic:**
   \[
   t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}
   \]

5. **Compare the t-statistic to the critical value of \(t\) for a one-tailed test or compute as np
from scipy.stats import t

# Data
data = [53, 57, 52, 65, 70, 51, 55, 61, 49, 61, 54, 64, 44, 57, 55, 60, 53, 61, 51, 56]
n = len(data)
mu_0 = 59.3  # Population mean

# Sample statistics
sample_mean = np.mean(data)
sample_std = np.std(data, ddof=1)  # Use ddof=1 for sample standard deviation

# Test statistic (t)
t_stat = (sample_mean - mu_0) / (sample_std / np.sqrt(n))

# Degrees of freedom
df = n - 1

# P-value (one-tailed test)
p_value = t.cdf(t_stat, df)

sample_mean, sample_std, t_stat,1. **Sample Mean (\(\bar{x}\))**: \(56.45\)
2. **Sample Standard Deviation (\(s\))**: \(6.13\) (rounded to 2 decimal places)
3. **t-statistic**: \(-2.08\)
4. **p-value**: \(0.026\) (rounded to 3 decimal places)

---

### Interpretation

1. **Compare p-value to \(\alpha\):**
   - Since the p-value (\(0.026\)) is less than the significance level (\(\alpha = 0.05\)), we reject the null hypothesis (\(H_0\)).

2. **Conclusion:**
   - There is enough evidence to suggest that the new guidelines have reduced the average number of weekly hours worked by residents.

---

### c. p-value
The p-value for the test is \(0.026\).

---

### Summary
The test indicates that the new guidelines were effective in reducing the average weekly hours worked by residents, as the p-value is statistically significant.

Do you have questions or need further clarification? Here are five questions to consider:
1. How does the t-test differ from other statistical tests?
2. What assumptions are required for this t-test to be valid?
3. How does the sample size impact the reliability of this test?
4. Could we perform a similar test if we had a larger dataset?
5. Why do we use a one-tailed test here instead of a two-tailed test?

**Tip**: Always verify the assumptions of normality when performing t-tests,

A survey found that in 2008, doctors in residency programs at U.S. hospitals worked an average of 59.3 hours per week. A new residency program guideline went into effect in 2009, with the aim of reducing the number of hours residents work per week. In 2009, a small survey of 20 residents found the following weekly hours worked: 53, 57, 52, 65, 70, 51, 55, 61, 49, 61, 54, 64, 44, 57, 55, 60, 53, 61, 51, 56. You would like to show that the new guidelines have reduced the number of hours worked per week. Perform a test using alpha = 0.05, and answer the following: a. State the Null and Alternative hypotheses b. Does the sample provide enough evidence to suggest that the new guidelines have reduced the number of hours worked per week? c. For the test in (b), what is the p-value?

Explore a hypothesis test to evaluate if new residency guidelines reduced weekly work hours for doctors. Analyze a dataset of 20 residents from 2009 using a one-sample t-test with alpha = 0.05. Detailed steps include hypothesis formulation, t-statistic calculation, and p-value interpretation.

Math Problem Statement

5 6 You would like to show that that the new guidelines have reduced the number of hours worked per week. Perform a test using alpha

0 . 0 5 , and answer the following: a . State the Null and Alternative hypotheses b . Does the sample provide enough evidence to suggest that the new guidelines have reduced the number of hours worked per week? c . For the test in ( b ) , what is the p

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