diff options
Diffstat (limited to 'numpy')
| -rw-r--r-- | numpy/random/_generator.pyx | 42 | ||||
| -rw-r--r-- | numpy/random/mtrand.pyx | 42 |
2 files changed, 54 insertions, 30 deletions
diff --git a/numpy/random/_generator.pyx b/numpy/random/_generator.pyx index a7d98e2ed..1903dce37 100644 --- a/numpy/random/_generator.pyx +++ b/numpy/random/_generator.pyx @@ -1732,33 +1732,45 @@ cdef class Generator: ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended - value of 7725 kJ? + value of 7725 kJ? Our null hypothesis will be the absence of deviation, + and the alternate hypothesis will be the presence of an effect that could be + either positive or negative, hence making our test 2-tailed. + + Because we are estimating the mean and we have N=11 values in our sample, + we have N-1=10 degrees of freedom. We set our signifance level to 95% and + compute the t statistic using the empirical mean and empirical standard + deviation of our intake. We use a ddof of 1 to base the computation of our + empirical standard deviation on an unbiased estimate of the variance (note: + the final estimate is not unbiased due to the concave nature of the square + root). - We have 10 degrees of freedom, so is the sample mean within 95% of the - recommended value? - - >>> s = np.random.default_rng().standard_t(10, size=100000) >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 + >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) + >>> t + -2.8207540608310198 - Calculate the t statistic, setting the ddof parameter to the unbiased - value so the divisor in the standard deviation will be degrees of - freedom, N-1. + We draw 1000000 samples from Student's t distribution with the adequate + degrees of freedom. - >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> import matplotlib.pyplot as plt + >>> s = np.random.default_rng().standard_t(10, size=1000000) >>> h = plt.hist(s, bins=100, density=True) - For a one-sided t-test, how far out in the distribution does the t - statistic appear? + Does our t statistic land in one of the two critical regions found at + both tails of the distribution? + + >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s)) + 0.018318 #random < 0.05, statistic is in critical region - >>> np.sum(s<t) / float(len(s)) - 0.0090699999999999999 # random + The probability value for this 2-tailed test is about 1.83%, which is + lower than the 5% pre-determined significance threshold. - So the p-value is about 0.009, which says the null hypothesis has a - probability of about 99% of being true. + Therefore, the probability of observing values as extreme as our intake + conditionally on the null hypothesis being true is too low, and we reject + the null hypothesis of no deviation. """ return cont(&random_standard_t, &self._bitgen, size, self.lock, 1, diff --git a/numpy/random/mtrand.pyx b/numpy/random/mtrand.pyx index df8d7e380..683705a05 100644 --- a/numpy/random/mtrand.pyx +++ b/numpy/random/mtrand.pyx @@ -2147,33 +2147,45 @@ cdef class RandomState: ... 7515, 8230, 8770]) Does their energy intake deviate systematically from the recommended - value of 7725 kJ? + value of 7725 kJ? Our null hypothesis will be the absence of deviation, + and the alternate hypothesis will be the presence of an effect that could be + either positive or negative, hence making our test 2-tailed. + + Because we are estimating the mean and we have N=11 values in our sample, + we have N-1=10 degrees of freedom. We set our signifance level to 95% and + compute the t statistic using the empirical mean and empirical standard + deviation of our intake. We use a ddof of 1 to base the computation of our + empirical standard deviation on an unbiased estimate of the variance (note: + the final estimate is not unbiased due to the concave nature of the square + root). - We have 10 degrees of freedom, so is the sample mean within 95% of the - recommended value? - - >>> s = np.random.standard_t(10, size=100000) >>> np.mean(intake) 6753.636363636364 >>> intake.std(ddof=1) 1142.1232221373727 + >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) + >>> t + -2.8207540608310198 - Calculate the t statistic, setting the ddof parameter to the unbiased - value so the divisor in the standard deviation will be degrees of - freedom, N-1. + We draw 1000000 samples from Student's t distribution with the adequate + degrees of freedom. - >>> t = (np.mean(intake)-7725)/(intake.std(ddof=1)/np.sqrt(len(intake))) >>> import matplotlib.pyplot as plt + >>> s = np.random.standard_t(10, size=1000000) >>> h = plt.hist(s, bins=100, density=True) - For a one-sided t-test, how far out in the distribution does the t - statistic appear? + Does our t statistic land in one of the two critical regions found at + both tails of the distribution? + + >>> np.sum(np.abs(t) < np.abs(s)) / float(len(s)) + 0.018318 #random < 0.05, statistic is in critical region - >>> np.sum(s<t) / float(len(s)) - 0.0090699999999999999 #random + The probability value for this 2-tailed test is about 1.83%, which is + lower than the 5% pre-determined significance threshold. - So the p-value is about 0.009, which says the null hypothesis has a - probability of about 99% of being true. + Therefore, the probability of observing values as extreme as our intake + conditionally on the null hypothesis being true is too low, and we reject + the null hypothesis of no deviation. """ return cont(&legacy_standard_t, &self._aug_state, size, self.lock, 1, |