Jacob.Fritsche/Hypothesentest_IU
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Jacob.Fritsche committed 2025-02-02 16:50:05 +00:00
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Tutorium 2.ipynb
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@@ -12,7 +12,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 25, "execution_count": 2,
"id": "a4ea5fe4-eaa9-4cc4-8677-385a68577332", "id": "a4ea5fe4-eaa9-4cc4-8677-385a68577332",
"metadata": {}, "metadata": {},
"outputs": [], "outputs": [],
@@ -30,15 +30,23 @@
"Generate a normal distribution with sample of an experiment" "Generate a normal distribution with sample of an experiment"
] ]
}, },
{
"cell_type": "markdown",
"id": "ed78c89c-0bee-4b52-be93-31a2698112ad",
"metadata": {},
"source": [
"Sample of people and measure their heights. Re run the code...also the average is changing."
]
},
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 35, "execution_count": 7,
"id": "280f78b3-4338-49ec-8929-732d429762db", "id": "280f78b3-4338-49ec-8929-732d429762db",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"data": { "data": {
"image/png": 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truncated "image/png": 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truncated
"text/plain": [ "text/plain": [
"<Figure size 640x480 with 1 Axes>" "<Figure size 640x480 with 1 Axes>"
] ]
@@ -49,7 +57,7 @@
], ],
"source": [ "source": [
"# Generate 1000 random numbers from a normal distribution\n", "# Generate 1000 random numbers from a normal distribution\n",
"mu, sigma = 3, 6 # mean and standard deviation\n", "mu, sigma = 180, 10 # mean and standard deviation\n",
"data = np.random.normal(mu, sigma, 1000)\n", "data = np.random.normal(mu, sigma, 1000)\n",
"mean_sample = data.mean().item()\n", "mean_sample = data.mean().item()\n",
"# Create a histogram of the data\n", "# Create a histogram of the data\n",
@@ -61,15 +69,23 @@
"plt.show()" "plt.show()"
] ]
}, },
{
"cell_type": "markdown",
"id": "82712fee-3e30-4e03-8be5-636d4193fb50",
"metadata": {},
"source": [
"As we can see, it seems to be appropriate, to approx this with the normal distribution"
]
},
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 29, "execution_count": 8,
"id": "0df9c2e5-ae6d-48c0-93a4-12a6e70240c2", "id": "0df9c2e5-ae6d-48c0-93a4-12a6e70240c2",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"data": { "data": {
"image/png": 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truncated "image/png": "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 truncated
"text/plain": [ "text/plain": [
"<Figure size 640x480 with 1 Axes>" "<Figure size 640x480 with 1 Axes>"
] ]
@@ -92,21 +108,79 @@
}, },
{ {
"cell_type": "markdown", "cell_type": "markdown",
"id": "f3c0c504-e041-4d85-8da0-3ef766f8e87a", "id": "967b597a-6602-49dc-882f-8a6ae69ccba8",
"metadata": {}, "metadata": {},
"source": [ "source": [
"Generate for the distribution of mu_sample" "Generate the distribution for mu_sample"
]
},
{
"cell_type": "markdown",
"id": "5533817e-90ec-4671-b1ec-196916c079e2",
"metadata": {},
"source": [
"Now we don't are interested in the height of the person, now we are looking the average. So we multiple generate samples of people and then we are calculating the average. Also teh average should be normal distrubited. Re run the code at above."
] ]
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 36, "execution_count": 9,
"id": "43be8991-a5b0-4f0f-be9c-0f86b77d96a7",
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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truncated
"text/plain": [
"<Figure size 640x480 with 1 Axes>"
]
},
"metadata": {},
"output_type": "display_data"
}
],
"source": [
"# Show the approximation with the normal distribution\n",
"mu, sigma = 180, 0.5 # mean and standard deviation\n",
"data = np.random.normal(mu, sigma, 1000)\n",
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
"xmin, xmax = plt.xlim()\n",
"x = np.linspace(xmin, xmax, 100)\n",
"p = stats.norm.pdf(x, mu, sigma)\n",
"plt.plot(x, p, 'k', linewidth=2)\n",
"title = \"Fit results: mu = %.2f, std = %.2f\" % (mu, sigma)\n",
"plt.title(title)\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"id": "f3c0c504-e041-4d85-8da0-3ef766f8e87a",
"metadata": {},
"source": [
"Generate the distribution of mu_sample and theoretical distribtion"
]
},
{
"cell_type": "markdown",
"id": "db5f4ad9-c37a-40b2-b815-6b1c34769f69",
"metadata": {},
"source": [
" <h1>Now one-way t-test</h1>\n",
" <p>We are assuming that the average height in the population is 178 cm, for example, in a sample from 1990. We want to prove that nowadays people are getting taller.</p>\n",
" <p><strong>H<sub>0</sub></strong> = 178 cm and <strong>H<sub>A</sub></strong> > 178 cm</p>\n",
" <p>So we have a one-sided test.</p>"
]
},
{
"cell_type": "code",
"execution_count": 17,
"id": "f7118230-fc88-4a71-b813-2daa0c63c97c", "id": "f7118230-fc88-4a71-b813-2daa0c63c97c",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"data": { "data": {
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truncated
"text/plain": [ "text/plain": [
"<Figure size 640x480 with 1 Axes>" "<Figure size 640x480 with 1 Axes>"
] ]
@@ -117,26 +191,26 @@
], ],
"source": [ "source": [
"# Generate 1000 random numbers from the first normal distribution\n", "# Generate 1000 random numbers from the first normal distribution\n",
"mu1, sigma1 = 3, 0.5 # mean and standard deviation for the first normal distribution\n", "mu1, sigma1 = 180, 0.5 # mean and standard deviation for the first normal distribution\n",
"data = np.random.normal(mu1, sigma1, 1000)\n", "data = np.random.normal(mu1, sigma1, 1000)\n",
"\n", "\n",
"# Create a histogram of the data\n", "# Create a histogram of the data\n",
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n", "plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
"\n", "\n",
"# Add a graph of the first normal distribution used to generate the data\n", "# Add a graph of the first normal distribution used to generate the data\n",
"xmin, xmax = -1, 5\n", "xmin, xmax = 175, 182\n",
"x = np.linspace(xmin, xmax, 100)\n", "x = np.linspace(xmin, xmax, 100)\n",
"p1 = stats.norm.pdf(x, mu1, sigma1)\n", "p1 = stats.norm.pdf(x, mu1, sigma1)\n",
"plt.plot(x, p1, 'k', linewidth=2)\n", "plt.plot(x, p1, 'k', linewidth=2)\n",
"\n", "\n",
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n", "# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
"mu2, sigma2 = 1, 0.4\n", "mu2, sigma2 = 178, 0.5\n",
"p2 = stats.norm.pdf(x, mu2, sigma2)\n", "p2 = stats.norm.pdf(x, mu2, sigma2)\n",
"plt.plot(x, p2, 'r', linewidth=2)\n", "plt.plot(x, p2, 'r', linewidth=2)\n",
"\n", "\n",
"# Add vertical lines at x = 1 and x = 3 with labels\n", "# Add vertical lines at x = 1 and x = 3 with labels\n",
"plt.axvline(x=1, color='b', linestyle='--', label='mu theoretical distribution')\n", "plt.axvline(x=178, color='b', linestyle='--', label='mu theoretical distribution')\n",
"plt.axvline(x=3, color='m', linestyle='--', label='mu in sample')\n", "plt.axvline(x=180, color='m', linestyle='--', label='mu in sample')\n",
"\n", "\n",
"title = \"Fit results: mu1 = %.2f, std1 = %.2f; mu2 = %.2f, std2 = %.2f\" % (mu1, sigma1, mu2, sigma2)\n", "title = \"Fit results: mu1 = %.2f, std1 = %.2f; mu2 = %.2f, std2 = %.2f\" % (mu1, sigma1, mu2, sigma2)\n",
"plt.title(title)\n", "plt.title(title)\n",
@@ -149,13 +223,13 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 37, "execution_count": 21,
"id": "8b4766b1-6fb5-487b-b552-790f43f83707", "id": "8b4766b1-6fb5-487b-b552-790f43f83707",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"data": { "data": {
"image/png": 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truncated
"text/plain": [ "text/plain": [
"<Figure size 640x480 with 1 Axes>" "<Figure size 640x480 with 1 Axes>"
] ]
@@ -166,26 +240,26 @@
], ],
"source": [ "source": [
"# Generate 1000 random numbers from the first normal distribution\n", "# Generate 1000 random numbers from the first normal distribution\n",
"mu1, sigma1 = 3, 0.5 # mean and standard deviation for the first normal distribution\n", "mu1, sigma1 = 180, 0.5 # mean and standard deviation for the first normal distribution\n",
"data = np.random.normal(mu1, sigma1, 1000)\n", "data = np.random.normal(mu1, sigma1, 1000)\n",
"\n", "\n",
"# Create a histogram of the data\n", "# Create a histogram of the data\n",
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n", "plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
"\n", "\n",
"# Add a graph of the first normal distribution used to generate the data\n", "# Add a graph of the first normal distribution used to generate the data\n",
"xmin, xmax = -1, 5\n", "xmin, xmax = 175, 182\n",
"x = np.linspace(xmin, xmax, 100)\n", "x = np.linspace(xmin, xmax, 100)\n",
"p1 = stats.norm.pdf(x, mu1, sigma1)\n", "p1 = stats.norm.pdf(x, mu1, sigma1)\n",
"plt.plot(x, p1, 'k', linewidth=2)\n", "plt.plot(x, p1, 'k', linewidth=2)\n",
"\n", "\n",
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n", "# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
"mu2, sigma2 = 1, 0.4\n", "mu2, sigma2 = 178, 0.5\n",
"p2 = stats.norm.pdf(x, mu2, sigma2)\n", "p2 = stats.norm.pdf(x, mu2, sigma2)\n",
"plt.plot(x, p2, 'r', linewidth=2)\n", "plt.plot(x, p2, 'r', linewidth=2)\n",
"\n", "\n",
"# Add vertical lines at x = 1 and x = 3 with labels\n", "# Add vertical lines at x = 1 and x = 3 with labels\n",
"plt.axvline(x=1, color='b', linestyle='--', label='mu theoretical distribution')\n", "plt.axvline(x=178, color='b', linestyle='--', label='mu theoretical distribution')\n",
"plt.axvline(x=3, color='m', linestyle='--', label='mu in sample')\n", "plt.axvline(x=180, color='m', linestyle='--', label='mu in sample')\n",
"\n", "\n",
"# Calculate the 95% quantile for the second normal distribution\n", "# Calculate the 95% quantile for the second normal distribution\n",
"quantile_95 = stats.norm.ppf(0.95, mu2, sigma2)\n", "quantile_95 = stats.norm.ppf(0.95, mu2, sigma2)\n",
@@ -206,7 +280,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 38, "execution_count": 22,
"id": "665a50fe-8404-4db3-b412-aec36d81016e", "id": "665a50fe-8404-4db3-b412-aec36d81016e",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
@@ -214,15 +288,15 @@
"name": "stdout", "name": "stdout",
"output_type": "stream", "output_type": "stream",
"text": [ "text": [
"T-statistic: 98.77295966495895, P-value: 0.0\n", "T-statistic: 89.44271909999159, P-value: 0.0\n",
"The means of the two theoretical distributions are statistically different (reject H0).\n" "The means of the two theoretical distributions are statistically different (reject H0).\n"
] ]
} }
], ],
"source": [ "source": [
"# Define the parameters for the two normal distributions\n", "# Define the parameters for the two normal distributions\n",
"mu1, sigma1 = 3, 0.5 # mean and standard deviation for the first normal distribution\n", "mu1, sigma1 = 180, 0.5 # mean and standard deviation for the first normal distribution\n",
"mu2, sigma2 = 1, 0.4 # mean and standard deviation for the second normal distribution\n", "mu2, sigma2 = 178, 0.5 # mean and standard deviation for the second normal distribution\n",
"\n", "\n",
"# Perform a t-test to compare the means of the two theoretical distributions\n", "# Perform a t-test to compare the means of the two theoretical distributions\n",
"# Since we don't have samples, we use the means and standard deviations directly\n", "# Since we don't have samples, we use the means and standard deviations directly\n",
@@ -254,13 +328,13 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 25, "execution_count": 23,
"id": "aa9ea3d5-3559-4fd7-a2c7-f041486c090e", "id": "aa9ea3d5-3559-4fd7-a2c7-f041486c090e",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"data": { "data": {
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truncated
"text/plain": [ "text/plain": [
"<Figure size 640x480 with 1 Axes>" "<Figure size 640x480 with 1 Axes>"
] ]
@@ -271,20 +345,20 @@
], ],
"source": [ "source": [
"# Generate 1000 random numbers from the first normal distribution\n", "# Generate 1000 random numbers from the first normal distribution\n",
"mu1, sigma1 = 1.8, 0.5 # mean and standard deviation for the first normal distribution\n", "mu1, sigma1 = 180, 0.5 # mean and standard deviation for the first normal distribution\n",
"data = np.random.normal(mu1, sigma1, 1000)\n", "data = np.random.normal(mu1, sigma1, 1000)\n",
"\n", "\n",
"# Create a histogram of the data\n", "# Create a histogram of the data\n",
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n", "plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
"\n", "\n",
"# Add a graph of the first normal distribution used to generate the data\n", "# Add a graph of the first normal distribution used to generate the data\n",
"xmin, xmax = -1, 5\n", "xmin, xmax = 175, 182\n",
"x = np.linspace(xmin, xmax, 100)\n", "x = np.linspace(xmin, xmax, 100)\n",
"p1 = stats.norm.pdf(x, mu1, sigma1)\n", "p1 = stats.norm.pdf(x, mu1, sigma1)\n",
"plt.plot(x, p1, 'k', linewidth=2)\n", "plt.plot(x, p1, 'k', linewidth=2)\n",
"\n", "\n",
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n", "# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
"mu2, sigma2 = 1, 0.4\n", "mu2, sigma2 = 179, 0.5\n",
"p2 = stats.norm.pdf(x, mu2, sigma2)\n", "p2 = stats.norm.pdf(x, mu2, sigma2)\n",
"plt.plot(x, p2, 'r', linewidth=2)\n", "plt.plot(x, p2, 'r', linewidth=2)\n",
"\n", "\n",
@@ -311,7 +385,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 39, "execution_count": 24,
"id": "0db5895c-28ab-4617-ad67-7ad8311d0341", "id": "0db5895c-28ab-4617-ad67-7ad8311d0341",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
@@ -319,15 +393,15 @@
"name": "stdout", "name": "stdout",
"output_type": "stream", "output_type": "stream",
"text": [ "text": [
"T-statistic: 39.50918386598358, P-value: 8.942863909656787e-253\n", "T-statistic: 44.721359549995796, P-value: 2.8557430532137358e-303\n",
"The means of the two theoretical distributions are statistically different (reject H0).\n" "The means of the two theoretical distributions are statistically different (reject H0).\n"
] ]
} }
], ],
"source": [ "source": [
"# Define the parameters for the two normal distributions\n", "# Define the parameters for the two normal distributions\n",
"mu1, sigma1 = 1.8, 0.5 # mean and standard deviation for the first normal distribution\n", "mu1, sigma1 = 180, 0.5 # mean and standard deviation for the first normal distribution\n",
"mu2, sigma2 = 1, 0.4 # mean and standard deviation for the second normal distribution\n", "mu2, sigma2 = 179, 0.5 # mean and standard deviation for the second normal distribution\n",
"\n", "\n",
"# Perform a t-test to compare the means of the two theoretical distributions\n", "# Perform a t-test to compare the means of the two theoretical distributions\n",
"# Since we don't have samples, we use the means and standard deviations directly\n", "# Since we don't have samples, we use the means and standard deviations directly\n",
@@ -351,13 +425,13 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 51, "execution_count": 25,
"id": "ab987426-9a37-4c22-b561-d2870b856071", "id": "ab987426-9a37-4c22-b561-d2870b856071",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
{ {
"data": { "data": {
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truncated
"text/plain": [ "text/plain": [
"<Figure size 640x480 with 1 Axes>" "<Figure size 640x480 with 1 Axes>"
] ]
@@ -368,20 +442,20 @@
], ],
"source": [ "source": [
"# Generate 1000 random numbers from the first normal distribution\n", "# Generate 1000 random numbers from the first normal distribution\n",
"mu1, sigma1 = 1.03, 0.5 # mean and standard deviation for the first normal distribution\n", "mu1, sigma1 = 180, 0.5 # mean and standard deviation for the first normal distribution\n",
"data = np.random.normal(mu1, sigma1, 1000)\n", "data = np.random.normal(mu1, sigma1, 1000)\n",
"\n", "\n",
"# Create a histogram of the data\n", "# Create a histogram of the data\n",
"plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n", "plt.hist(data, bins=30, density=True, alpha=0.6, color='g')\n",
"\n", "\n",
"# Add a graph of the first normal distribution used to generate the data\n", "# Add a graph of the first normal distribution used to generate the data\n",
"xmin, xmax = -1, 5\n", "xmin, xmax = 175, 182\n",
"x = np.linspace(xmin, xmax, 100)\n", "x = np.linspace(xmin, xmax, 100)\n",
"p1 = stats.norm.pdf(x, mu1, sigma1)\n", "p1 = stats.norm.pdf(x, mu1, sigma1)\n",
"plt.plot(x, p1, 'k', linewidth=2)\n", "plt.plot(x, p1, 'k', linewidth=2)\n",
"\n", "\n",
"# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n", "# Add a graph of the second normal distribution with mu = 1 and sigma = 0.4\n",
"mu2, sigma2 = 1, 0.4\n", "mu2, sigma2 = 179.8, 0.5\n",
"p2 = stats.norm.pdf(x, mu2, sigma2)\n", "p2 = stats.norm.pdf(x, mu2, sigma2)\n",
"plt.plot(x, p2, 'r', linewidth=2)\n", "plt.plot(x, p2, 'r', linewidth=2)\n",
"\n", "\n",
@@ -408,7 +482,7 @@
}, },
{ {
"cell_type": "code", "cell_type": "code",
"execution_count": 50, "execution_count": 29,
"id": "9c019dea-d687-4dc1-b48f-fe629e54172e", "id": "9c019dea-d687-4dc1-b48f-fe629e54172e",
"metadata": {}, "metadata": {},
"outputs": [ "outputs": [
@@ -416,15 +490,15 @@
"name": "stdout", "name": "stdout",
"output_type": "stream", "output_type": "stream",
"text": [ "text": [
"T-statistic: 1.4815943949743855, P-value: 0.13860593668122914\n", "T-statistic: 8.944271909998651, P-value: 8.328998310440721e-19\n",
"The means of the two theoretical distributions are not statistically different (fail to reject H0).\n" "The means of the two theoretical distributions are not statistically different (fail to reject H0).\n"
] ]
} }
], ],
"source": [ "source": [
"# Define the parameters for the two normal distributions\n", "# Define the parameters for the two normal distributions\n",
"mu1, sigma1 = 1.03, 0.5 # mean and standard deviation for the first normal distribution\n", "mu1, sigma1 = 180, 0.5 # mean and standard deviation for the first normal distribution\n",
"mu2, sigma2 = 1, 0.4 # mean and standard deviation for the second normal distribution\n", "mu2, sigma2 = 179.8, 0.5 # mean and standard deviation for the second normal distribution\n",
"\n", "\n",
"# Perform a t-test to compare the means of the two theoretical distributions\n", "# Perform a t-test to compare the means of the two theoretical distributions\n",
"# Since we don't have samples, we use the means and standard deviations directly\n", "# Since we don't have samples, we use the means and standard deviations directly\n",
@@ -439,7 +513,7 @@
"print(f\"T-statistic: {t_statistic}, P-value: {p_value}\")\n", "print(f\"T-statistic: {t_statistic}, P-value: {p_value}\")\n",
"\n", "\n",
"# Interpret the result\n", "# Interpret the result\n",
"alpha = 0.05\n", "alpha = 5e-20\n",
"if p_value < alpha:\n", "if p_value < alpha:\n",
" print(\"The means of the two theoretical distributions are statistically different (reject H0).\")\n", " print(\"The means of the two theoretical distributions are statistically different (reject H0).\")\n",
"else:\n", "else:\n",