جواب تمرین صفحه 68 درس 3 ریاضی و آمار یازدهم انسانی (آمار)

الف سری زمانی است. فواصل نوسانی منظم است.

ب سری زمانی نیست. فواصل نوسانی منظم بیان نشده است.

پ سری زمانی نیست. فواصل نوسانی منظم بیان نشده است.

شکل‌های الف و ب، چون تقریباً بر روی یک خط تغییر می‌کنند و تغییرات ناگهانی ندارند.

الف

ب

\(\begin{array}{l}m = \frac{{25 - 8}}{{7 - 1}} = \frac{{17}}{6}\\\\\frac{{17}}{6}(1) + h = 8\\\\ \Rightarrow h = 8 - \frac{{17}}{6} = \frac{{48 - 17}}{6} = \frac{{31}}{6}\\\\f(x) = \frac{{17}}{6}x + \frac{{21}}{6}\\\\f(2) = \frac{{17}}{6}(2) + \frac{{21}}{6} = \frac{{34 + 21}}{6} = \frac{{55}}{6}\\\\f(4) = \frac{{17}}{6}(4) + \frac{{21}}{6} = \frac{{68 + 21}}{6} = \frac{{89}}{6}\\\\f(6) = \frac{{17}}{6}(6) + \frac{{21}}{6} = \frac{{102 + 21}}{6} = \frac{{123}}{6} = 20/5\end{array}\)

پ

\(\begin{array}{l}\bar x = \frac{{1 + 3 + 5 + 7}}{4} = \frac{{16}}{4} = 4\\\\\bar y = \frac{{8 + 13 + 16 + 25}}{4} = \frac{{62}}{4} = 15/5\\\\\frac{{19}}{6}(7) + h = 25 \Rightarrow h = 25 - \frac{{133}}{6} = \frac{{150 - 133}}{6} = \frac{{17}}{6}\\\\f(x) = \frac{{19}}{6}x + \frac{{17}}{6}\\\\ \Rightarrow f(10) = \frac{{19}}{6}(10) + \frac{{17}}{6} = \frac{{190 + 17}}{6} = \frac{{207}}{6} = 34/5\end{array}\)

تعداد بطری های دوشنبه هفته دوم

\(m = \frac{{25 - 15/5}}{{7 - 4}} = \frac{{9/5}}{3} = \frac{{9/5}}{{30}} = \frac{{95}}{{30}} = \frac{{19}}{6}\)

\(\begin{array}{l}\bar x = \frac{{20 + 22 + 24 + 26 + 28}}{5} = \frac{{120}}{5} = 24\\\\\bar y = \frac{{30 + 27 + 32 + 35 + 35}}{5} = \frac{{159}}{5} = 31/8\\\\m = \frac{{35 - 31/8}}{{28 - 24}} = \frac{{3/2}}{4} = \frac{{32}}{{40}} = \frac{4}{5}\\\\\frac{4}{5}(28) + h = 35 \Rightarrow h = - \frac{{112}}{5} + 35 = \frac{{ - 112 + 175}}{5} = \frac{{63}}{5}\\\\f(x) = \frac{4}{5}x + \frac{{63}}{5}\\\\f(30) = \frac{4}{5}(30) + \frac{{63}}{5} = \frac{{120 + 63}}{5} = \frac{{183}}{5} = 36/6\end{array}\)

الف

ب

\(\begin{array}{l}\bar x = \frac{{1 + 2 + 3 + 4 + 5 + 6}}{6} = \frac{{21}}{6}\\\\\bar y = \frac{{19/5 + 29 + 38/5 + 48 + 58 + 67}}{6} = \frac{{260}}{6}\\\\m = \frac{{67 - \frac{{260}}{6}}}{{6 - \frac{{21}}{6}}} = \frac{{\frac{{402 - 260}}{6}}}{{\frac{{36 - 21}}{6}}} = \frac{{142}}{{15}}\\\\\frac{{142}}{{15}}(6) + h = 67 \Rightarrow h = - \frac{{852}}{{15}} + 35 = \frac{{ - 852 + 1005}}{{15}} = \frac{{153}}{{15}}\\\\(x) = \frac{{142}}{{15}}(x) + \frac{{153}}{{15}}\\\\f(7) = \frac{{14/2}}{{15}}(7) + \frac{{153}}{{15}} = \frac{{994 + 153}}{{15}} = \frac{{1147}}{{15}} = 76/47\end{array}\)

الف

ب

\(\begin{array}{l}\bar x = \frac{{1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10}}{{10}} = \frac{{55}}{{10}} = 5/5\\\\\bar y = \frac{{30 + 28 + 29 + 23 + 20 + 16 + 21 + 25 + 16 + 21}}{{10}} = \frac{{229}}{{10}} = 22/9\end{array}\)

پ

\(\begin{array}{l}m = \frac{{21 - 22/9}}{{10 - 5/5}} = \frac{{ - 1/9}}{{4/5}} = - \frac{{19}}{{45}}\\\\ - \frac{{19}}{{45}}(10) + h = 21 \Rightarrow h = 21 + \frac{{190}}{{45}} \Rightarrow h = \frac{{945 + 190}}{{45}} = \frac{{1135}}{{45}}\\\\ \Rightarrow f(x) = - \frac{{19}}{{45}}x + \frac{{1135}}{{45}}\end{array}\)

ت

\(f(11) = - \frac{{19}}{{45}}(11) + \frac{{1135}}{{45}} = \frac{{ - 209 + 1135}}{{45}} = \frac{{926}}{{45}} = 20/58\)

ث

خطای برون یابی:

\(E = |20/58 - 25| = 4/42\)

الف

ب

توجه: سال ۱۳۱۶ را ۱ در نظر گرفته‌ایم، بنابراین سال ۱۳۸۶ سال ۷۱ است.

\(\begin{array}{l}m = \frac{{18/4 - 21/2}}{{71 - 1}} = \frac{{ - 2/8}}{{70}} = - \frac{4}{{100}} - \frac{4}{{100}}(71) + h = 18/4\\\\ \Rightarrow h = \frac{{284}}{{100}} + \frac{{1840}}{{100}} \Rightarrow h = \frac{{2124}}{{100}}\\\\ \Rightarrow f(x) = - \frac{4}{{100}}x + \frac{{2124}}{{100}}\end{array}\)

پ

در سال 1326

\(\begin{array}{l} \Rightarrow f(11) = - \frac{4}{{100}}(11) + \frac{{2124}}{{100}} = \frac{{ - 44 + 2124}}{{100}} = \frac{{2080}}{{100}} = 20/8\\\\E = |20/8 - 6/6| = 14/2\end{array}\)

در سال 1331

\(\begin{array}{l} \Rightarrow f(16) = - \frac{4}{{100}}(16) + \frac{{2124}}{{100}} = \frac{{ - 64 + 2124}}{{100}} = \frac{{2060}}{{100}} = 20/6\\\\E = |20/6 - 7/2| = 13/4\end{array}\)

در سال 1336

\(\begin{array}{l} \Rightarrow f(21) = - \frac{4}{{100}}(21) + \frac{{2124}}{{100}} = \frac{{ - 84 + 2124}}{{100}} = \frac{{2040}}{{100}} = 20/4\\\\E = |20/4 - 4/4| = 16\end{array}\)

در سال 1341

\(\begin{array}{l} \Rightarrow f(26) = - \frac{4}{{100}}(26) + \frac{{2124}}{{100}} = \frac{{ - 104 + 2124}}{{100}} = \frac{{2020}}{{100}} = 20/2\\\\E = |20/2 - 0/9| = 19/3\end{array}\)

در سال 1346

\(\begin{array}{l} \Rightarrow f(31) = - \frac{4}{{100}}(31) + \frac{{2124}}{{100}} = \frac{{ - 124 + 2124}}{{100}} = \frac{{2000}}{{100}} = 20\\\\E = |20 - 0/8| = 19/2\end{array}\)

در سال 1351

\(\begin{array}{l}f(36) = - \frac{4}{{100}}(36) + \frac{{2124}}{{100}} = \frac{{ - 144 + 2124}}{{100}} = \frac{{1980}}{{100}} = 19/8\\\\E = |19/8 - 7/2| = 12/6\end{array}\)

در سال 1356

\(\begin{array}{l}f(41) = - \frac{4}{{100}}(41) + \frac{{2124}}{{100}} = \frac{{ - 164 + 2124}}{{100}} = \frac{{1960}}{{100}} = 19/6\\\\E = |19/6 - 25/1| = 5/5\end{array}\)

در سال 1361

\(\begin{array}{l}f(46) = - \frac{4}{{100}}(46) + \frac{{2124}}{{100}} = \frac{{ - 184 + 2124}}{{100}} = \frac{{1940}}{{100}} = 19/4\\\\E = |19/4 - 19/2| = 0/2\end{array}\)

در سال 1366

\(\begin{array}{l}f(51) = - \frac{4}{{100}}(51) + \frac{{2124}}{{100}} = \frac{{ - 204 + 2124}}{{100}} = \frac{{1920}}{{100}} = 19/2\\\\E = |19/2 - 27/7| = 8/5\end{array}\)

در سال 1371

\(\begin{array}{l}f(56) = - \frac{4}{{100}}(56) + \frac{{2124}}{{100}} = \frac{{ - 224 + 2124}}{{100}} = \frac{{1900}}{{100}} = 19\\\\E = |19 - 24/4| = 5/4\end{array}\)

در سال 1376

\(\begin{array}{l}f(61) = - \frac{4}{{100}}(61) + \frac{{2124}}{{100}} = \frac{{ - 244 + 2124}}{{100}} = \frac{{1880}}{{100}} = 18/8\\\\E = |18/8 - 17/3| = 1/5\end{array}\)

در سال 1381

\(\begin{array}{l}f(66) = - \frac{4}{{100}}(66) + \frac{{2124}}{{100}} = \frac{{ - 264 + 2124}}{{100}} = \frac{{1860}}{{100}} = 18/6\\\\E = |18/6 - 15/8| = 2/8\end{array}\)

در سال 1386

\(\begin{array}{l}f(71) = - \frac{4}{{100}}(71) + \frac{{2124}}{{100}} = \frac{{ - 284 + 2124}}{{100}} = \frac{{1840}}{{100}} = 18/4\\\\E = |18/4 - 18/4| = 0\end{array}\)