In: Statistics and Probability
It is rare that you will find a gas station these days that only sells gas. It has become more common to find a convenient store that also sells gas. The data named “Convenient Shopping data” the sales over time at a franchise outlet of the major US oil company. Each row summarize sales for one day. This particular station sells gas and has a convenient store and car awash. The column labeled Sales gives the dollar sales of the convenient store and the column Volume gives the number of gallons of gas sold.
Sales (Dollars) | Volume (Gallons) |
1756 | 2933 |
2203 | 3329 |
1848 | 3043 |
2016 | 3043 |
2346 | 3450 |
2410 | 3478 |
2050 | 3347 |
2097 | 3708 |
2311 | 3467 |
2419 | 4114 |
2523 | 3721 |
2061 | 3448 |
2247 | 3230 |
3479 | 3557 |
2135 | 3060 |
2102 | 3619 |
2536 | 3256 |
1227 | 1757 |
1966 | 2891 |
2219 | 3381 |
2226 | 2970 |
1969 | 3301 |
2044 | 3178 |
2360 | 3426 |
1907 | 3118 |
2156 | 3037 |
1816 | 3537 |
1897 | 3808 |
2051 | 3145 |
2079 | 3766 |
2328 | 2916 |
1841 | 3957 |
2104 | 3980 |
1973 | 3675 |
2089 | 3516 |
2266 | 4149 |
2327 | 3733 |
2032 | 3738 |
2137 | 4012 |
2186 | 4114 |
2369 | 3795 |
2087 | 3543 |
2273 | 3681 |
2113 | 3618 |
2181 | 4452 |
2776 | 4346 |
2652 | 4073 |
2250 | 4260 |
2548 | 4113 |
2678 | 3829 |
2878 | 4137 |
2220 | 4269 |
2303 | 3989 |
2718 | 4238 |
2317 | 3658 |
2338 | 4005 |
2143 | 3996 |
2402 | 4077 |
2401 | 3610 |
2051 | 3701 |
2468 | 3844 |
2398 | 3904 |
2106 | 3879 |
2461 | 3266 |
2466 | 3513 |
2745 | 4052 |
1994 | 4052 |
2020 | 2874 |
2241 | 3526 |
2648 | 3487 |
2022 | 3499 |
2524 | 3236 |
1919 | 2422 |
2164 | 2876 |
2074 | 2883 |
2310 | 2771 |
2062 | 2362 |
1807 | 2564 |
1976 | 2708 |
2171 | 2519 |
1745 | 2638 |
2108 | 3448 |
2057 | 1993 |
1679 | 2560 |
2014 | 2777 |
2109 | 3097 |
2274 | 2750 |
2640 | 3260 |
1664 | 2050 |
1913 | 2921 |
2331 | 2970 |
1920 | 2624 |
2074 | 3496 |
2272 | 3729 |
1651 | 2302 |
1996 | 2672 |
2093 | 3150 |
1995 | 2948 |
2337 | 3520 |
2433 | 3195 |
1731 | 2232 |
2183 | 2979 |
1795 | 3178 |
1689 | 2618 |
2040 | 3117 |
2076 | 2847 |
1483 | 2150 |
930 | 1528 |
1674 | 2309 |
1934 | 2805 |
2011 | 2721 |
2172 | 2812 |
1612 | 2173 |
1780 | 2767 |
2116 | 2544 |
1937 | 2805 |
1866 | 2131 |
2099 | 3292 |
2082 | 2221 |
1788 | 2816 |
2004 | 2686 |
1868 | 3207 |
2038 | 2925 |
2596 | 3603 |
1700 | 2165 |
1815 | 3338 |
1917 | 3107 |
2143 | 2906 |
2420 | 3448 |
2486 | 3433 |
1812 | 2104 |
2463 | 3283 |
2222 | 3750 |
2324 | 3494 |
2219 | 3154 |
2505 | 3465 |
2047 | 2216 |
2231 | 3236 |
2067 | 3425 |
2293 | 3667 |
2152 | 3618 |
1366 | 2257 |
2210 | 3606 |
2029 | 3460 |
2742 | 2336 |
2161 | 3113 |
2223 | 3058 |
2186 | 2429 |
2306 | 3501 |
1933 | 3183 |
2485 | 3337 |
2817 | 3566 |
2491 | 3398 |
1896 | 2519 |
2382 | 3716 |
2552 | 3856 |
2094 | 3488 |
2447 | 3457 |
2440 | 3831 |
2041 | 2280 |
2261 | 2411 |
2114 | 3208 |
2866 | 3539 |
2752 | 3719 |
2502 | 4150 |
1786 | 2927 |
2157 | 3044 |
2025 | 3390 |
2327 | 3840 |
2502 | 3697 |
2552 | 4104 |
2017 | 3749 |
2019 | 3511 |
2302 | 3972 |
2419 | 3413 |
2921 | 3882 |
2273 | 3950 |
2183 | 3292 |
2428 | 3979 |
2489 | 4668 |
2037 | 3832 |
2324 | 3930 |
2591 | 3853 |
2362 | 4014 |
3001 | 4759 |
1801 | 2661 |
1744 | 4165 |
2428 | 4139 |
2409 | 3664 |
2819 | 3851 |
1897 | 2522 |
1536 | 1208 |
2475 | 3844 |
2484 | 3766 |
2117 | 3535 |
2488 | 3900 |
2553 | 3900 |
2251 | 3814 |
2435 | 3387 |
2446 | 4009 |
2063 | 1951 |
2582 | 3779 |
1663 | 2368 |
2302 | 3379 |
2248 | 3549 |
2712 | 3807 |
2307 | 4009 |
2576 | 3759 |
1978 | 2378 |
2116 | 4090 |
2292 | 3241 |
2373 | 3874 |
2444 | 4142 |
2578 | 3645 |
1953 | 2419 |
2151 | 3289 |
2901 | 3872 |
2514 | 4136 |
2078 | 3626 |
2492 | 4240 |
1897 | 2415 |
2072 | 3028 |
2538 | 3731 |
2422 | 3851 |
2415 | 3818 |
2969 | 4268 |
1775 | 2514 |
2082 | 3708 |
2121 | 3367 |
2471 | 3685 |
2467 | 3415 |
2671 | 4226 |
1876 | 2061 |
1976 | 3805 |
2156 | 3427 |
2339 | 3670 |
2258 | 3939 |
2776 | 3798 |
2084 | 2668 |
2346 | 3945 |
2320 | 3787 |
2539 | 3854 |
2393 | 3598 |
2629 | 3717 |
2044 | 2536 |
2018 | 401 |
2350 | 2361 |
2452 | 4005 |
2041 | 2391 |
2038 | 3129 |
2181 | 3874 |
2516 | 4072 |
2181 | 3603 |
2427 | 4173 |
2111 | 3993 |
2182 | 3153 |
2794 | 3812 |
1) Draw a scatter plot for Sales on Volume where Sales is dependent on Volume of gas sold. Does there appear to be a linear pattern that relates to these two sequences?
2) Estimate the linear regression model using excel analysis tool I showed you in class. Write the linear model and interpret the slope (b1).
3) Interpret the R2 and tell if your linear model is a good fit or not.
4) Estimate the difference in sales at the convenient store (on average) between a day with 3,500 gallons sold and a day with 4,000 gallons sold.
5) With regard to inference statistics, formulate a hypothesis test for the slope (b1) and decide if it is statistically significant or not.
6) Construct a 95% confidence interval for the slope.
1)
Yes,it appears be a linear pattern that relates to these two sequences
2)
SUMMARY OUTPUT | ||||||
Regression Statistics | ||||||
Multiple R | 0.6281 | |||||
R Square | 0.3945 | |||||
Adjusted R Square | 0.3921 | |||||
Standard Error | 249.5369 | |||||
Observations | 257 | |||||
ANOVA | ||||||
df | SS | MS | F | Significance F | ||
Regression | 1 | 10344773.5398 | 10344773.5398 | 166.1313 | 0.0000 | |
Residual | 255 | 15878506.9193 | 62268.6546 | |||
Total | 256 | 26223280.4591 | ||||
Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
Intercept | 1180.0870 | 81.6478 | 14.4534 | 0.0000 | 1019.2972 | 1340.8768 |
Volume (Gallons) | 0.3095 | 0.0240 | 12.8892 | 0.0000 | 0.2622 | 0.3568 |
y^ = 1180.087 + 0.3095 x
slope - when volume increase by 1 gallons, on average sales increase 0.3095 dollars
3)
R^2 = 0.3945
which means 39.45% of variation is explained by this model
4)
difference = 0.3095 * 500 = 154.75
5)
p-value for b1 = 0.0000
p-value < alpha
hence we reject the null hypothesis
it is statistically significant
6) 95% confidence interval for slope
(0.2622 , 0.3568)
0.2622 | 0.3568 |
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