Math Problem Statement

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?

Solution

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=20n = 20) and the population standard deviation is unknown. Let’s proceed step by step.


a. Null and Alternative Hypotheses

  1. Null Hypothesis (H0H_0): The average weekly hours worked under the new guidelines is the same as or more than before: H0:μ59.3H_0: \mu \geq 59.3
  2. Alternative Hypothesis (HaH_a): The average weekly hours worked under the new guidelines has been reduced: Ha:μ<59.3H_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 (nn) = 20
    • Sample data: 53,57,52,65,70,51,55,61,49,61,54,64,44,57,55,60,53,61,51,5653, 57, 52, 65, 70, 51, 55, 61, 49, 61, 54, 64, 44, 57, 55, 60, 53, 61, 51, 56
    • Population mean (μ0\mu_0) = 59.3
    • Significance level (α\alpha) = 0.05
  2. Calculate the sample mean (xˉ\bar{x}): xˉ=Sum of all sample valuesn\bar{x} = \frac{\text{Sum of all sample values}}{n}

  3. Calculate the sample standard deviation (ss): s=(xixˉ)2n1s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}

  4. Compute the t-statistic: t=xˉμ0s/nt = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}

  5. **Compare the t-statistic to the critical value of tt 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 (xˉ\bar{x}): 56.4556.45 2. Sample Standard Deviation (ss): 6.136.13 (rounded to 2 decimal places) 3. t-statistic: 2.08-2.08 4. p-value: 0.0260.026 (rounded to 3 decimal places)


Interpretation

  1. Compare p-value to α\alpha:

    • Since the p-value (0.0260.026) is less than the significance level (α=0.05\alpha = 0.05), we reject the null hypothesis (H0H_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 0.0260.026.


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,

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Math Problem Analysis

Mathematical Concepts

Hypothesis Testing
One-Sample t-Test
Statistical Significance

Formulas

Sample Mean: \(\bar{x} = \frac{\text{Sum of all sample values}}{n}\)
Sample Standard Deviation: \(s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}\)
t-Statistic: \(t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}}\)

Theorems

Central Limit Theorem
Student's t-Distribution

Suitable Grade Level

Grades 11-12, Undergraduate Statistics