![](\"https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook_2024/main/images/rowingTeam.png\")
\n",
"\n",
"Here are the heights in cm of eight men on a rowing team"
]
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\n",
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" height\n",
"count 8.000000\n",
"mean 180.412500\n",
"std 9.013868\n",
"min 166.500000\n",
"25% 176.500000\n",
"50% 180.050000\n",
"75% 182.250000\n",
"max 198.100000"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
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"source": [
"heights.describe()"
]
},
{
"cell_type": "markdown",
"id": "ad893005",
"metadata": {},
"source": [
"It looks like the mean height of rowers (180.4cm) is above the population mean of 175cm.\n",
"\n",
"Is this a big difference? It is less than one standard deviation above the mean: s=8.4 and (180.4-175)=5.4. \n",
"It turns out (from the normal distribution) that if the average rower was an actual person, they would be taller than about 75% of others, so tall, but not remarkably so. \n",
"\n",
"However, another key element is the variability in the dataset - if the rowers are consistently quite tall (that is most of them are tall), they may as a group be significantly taller than average."
]
},
{
"cell_type": "markdown",
"id": "5118a55a",
"metadata": {},
"source": [
"### Carry out the test\n",
"\n",
"To do the t-test itself we use a one-sample t-test using the function ttest_1samp (that's `stats.1samp()` for one sample) from \n",
"`scipy.stats`"
]
},
{
"cell_type": "code",
"execution_count": 103,
"id": "fc50b0c4",
"metadata": {},
"outputs": [
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"data": {
"text/plain": [
"Ttest_1sampResult(statistic=1.6983675966357161, pvalue=0.06662234508287802)"
]
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"execution_count": 103,
"metadata": {},
"output_type": "execute_result"
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"source": [
"stats.ttest_1samp(heights, 175, alternative='greater')"
]
},
{
"cell_type": "markdown",
"id": "56a0749d",
"metadata": {},
"source": [
"The inputs to stats.ttest_1samp are the sample (from our Pandas data frame heights) and the value to which they are to be compared (the average height of british men, 175cm),\n",
"and the argument alternative='greater', which tells the computer to run a one tailed test \n",
"that mean of the first input heights is greater than the reference value 175.\n",
"\n",
"The outputs are statistic ($t=1.70$) and pvalue ($p=0.067$) - if this is less than our $\\alpha$ value 0.5, there is a significant difference.\n",
"\n",
"### Degrees of freedom\n",
"\n",
"In a scientific write-up we also need to report the degrees of freedom of the test. This tells us how many observations (data-points) the test was based on, corrected for the number of means we had to estimate from the data in order to do the test.\n",
"\n",
"In the case of the one sample t-test $df = n-1$ so in this case, df=7 and we can report out test results as:\n",
"\n",
"$t(7) = 1.70, p=0.067$ (one-tailed)\n",
"\n",
"### Interpretation\n",
"\n",
"Our t value of 1.67 means that the difference between the mean height of the rowers and the standard value (175cm, the mean height of British men) is 1.67 times the standard error (where $ SE = \\frac{s}{\\sqrt{n}}$).\n",
"\n",
"Such a large difference (in the expected direction) would occur 0.067 (6.7%) of the time due to chance if the null hypothesis were true (rowers were no taller than other men), hence the p value of 0.067. This is higher than our alpha value (0.05) so the test is not significant.\n",
"\n",
"This diagram shows the expected distribution of t-values if the null were true, with our obtained t-value marked:\n",
"\n",
"| \n", " | height | \n", "
|---|---|
| count | \n", "8.000000 | \n", "
| mean | \n", "180.412500 | \n", "
| std | \n", "9.013868 | \n", "
| min | \n", "166.500000 | \n", "
| 25% | \n", "176.500000 | \n", "
| 50% | \n", "180.050000 | \n", "
| 75% | \n", "182.250000 | \n", "
| max | \n", "198.100000 | \n", "
![\"There](\"https://raw.githubusercontent.com/jillxoreilly/StatsCourseBook/main/images/ttestRowing.png\")
"
]
},
{
"cell_type": "markdown",
"id": "1746c632",
"metadata": {},
"source": [
"### Draw conclusions\n",
"\n",
"As p>0.05 the test is not significant and we fail to reject the null hypothesis. There is not sufficient evidence to support the hypothesis that rowers are taller than average."
]
},
{
"cell_type": "markdown",
"id": "2309a35d",
"metadata": {},
"source": [
"## Write-up \n",
"\n", "\n", "
\n",
" \n",
"Above, I walked you through how to run the t-test and why we make different choices. \n",
" \n",
"In this section we practice writing up our analysis in the correct style for a scientific report. \n",
" \n",
"Replace the XXXs with the correct values! \n",
"\n",
"
\n",
"\n",
"" ] }, { "cell_type": "markdown", "id": "eb2962b0", "metadata": {}, "source": [ "To test the hypothesis that rowers are taller than average, we measaured the heights of 8 male rowers and compared to the national value for mean height of British men (obtained from Wikipedia): 175cm.\n", "\n", "The data are shown below:" ] }, { "cell_type": "code", "execution_count": null, "id": "513809b5", "metadata": { "tags": [] }, "outputs": [], "source": [ "plot = sns.kdeplot(data = heights, fill=True)\n", "sns.rugplot(data = heights, height=0.1)\n", "plt.xlabel(\"height (cm)\", fontsize = 12)\n", "plt.ylabel(\"density\", fontsize = 12)\n", "plt.show()" ] }, { "cell_type": "markdown", "id": "4b88dc33", "metadata": {}, "source": [ "The mean height of the rowers was XXX.X cm and the standard deviation was X.XX cm." ] }, { "cell_type": "code", "execution_count": 14, "id": "2e3ee40e", "metadata": {}, "outputs": [ { "data": { "text/html": [ "
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" height\n",
"count 8.000000\n",
"mean 180.412500\n",
"std 9.013868\n",
"min 166.500000\n",
"25% 176.500000\n",
"50% 180.050000\n",
"75% 182.250000\n",
"max 198.100000"
]
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"execution_count": 14,
"metadata": {},
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"source": [
"heights = pandas.read_csv('data/rowersHeights.csv')\n",
"heights.describe()"
]
},
{
"cell_type": "markdown",
"id": "cac7ffd1",
"metadata": {},
"source": [
"We used a one-sample t-test (alpha = X, XX tailed). The use of the t-test (assumption of normality) was justified theoretically on the grounds that heights are typically normally distributed."
]
},
{
"cell_type": "code",
"execution_count": 15,
"id": "9b8270ad",
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Ttest_1sampResult(statistic=array([1.6983676]), pvalue=array([0.06662235]))"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
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"source": [
"stats.ttest_1samp(heights, 175, alternative='greater')"
]
},
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"cell_type": "markdown",
"id": "f8ad7dd7",
"metadata": {},
"source": [
"The results of the t-test $t(XX) =$ X.XX, $p=$X.XX (XX-tailed) suggest that there is insufficient evidence ot reject the null and we cannot conclude that rowers are taller than average."
]
},
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| \n", " | height | \n", "
|---|---|
| count | \n", "8.000000 | \n", "
| mean | \n", "180.412500 | \n", "
| std | \n", "9.013868 | \n", "
| min | \n", "166.500000 | \n", "
| 25% | \n", "176.500000 | \n", "
| 50% | \n", "180.050000 | \n", "
| 75% | \n", "182.250000 | \n", "
| max | \n", "198.100000 | \n", "