QNT Week 6

11

RUNNING HEAD: SIGNATURE ASSIGNMENT

Signature Assignment – Week 6

Part 1: Preliminary Analysis

This purpose of this study is to conclude the average census for hospitals in the US, along with determining the types of hospital ownership, proportions of general medical and psychiatric hospitals and to determine the number of births and average number of employed personnel in an average hospital in the US. The following questions will be answered in this study are as follows:

· What is the average census for an average hospital in the US?

· What are the main types of hospital ownerships in the US?

· What are the proportions of general medical and psychiatric hospital?

· What is the average number of births in an average hospital in the US?

· What is the average number of personnel employed by an average hospital in the US?

The sample in the study contains all general medical and psychiatric hospitals from seven selected areas within the United States. The seven areas that will be tested are California, South, Southeast, Northeast, Northwest, Midwest, and Rocky Mountain. The sample of the study consists of a total of 200 hospitals selected at random from the seven regions.

The geographical regions of the hospitals are a representation of qualitative data. Control similarly represents qualitative data, as it indicates the sort of ownership for each hospital. Similarly, service is a qualitative data and represents the type of the hospital, general medical or psychiatric. Census, births and personnel all signify quantitative data, as they contain numerical numbers. Geographical region, control and service data all are measured in the nominal level. This is due to the fact that they are simply names with no particular order. Census, births and personnel are all measured in the ratio level and all have a meaningful zero. (Hardy & Bryman, 2009).

Part 2: Descriptive Statistics

From the information on geographical region, the distributions of the hospitals in the seven areas are as follows; South- 56, Northwest- 30, Midwest- 60, Southwest- 3, Rocky Mountain- 20, California- 19 and Northwest-12. From the information on control, the ownership of the hospitals in the study is as follows; government, nonfederal- 51, non-government, not-for-profit- 86, for profit- 45 and federal government- 18. From the service data, 168 hospitals in the sample were general medical hospitals and 32 were psychiatric hospitals.

Microsoft Excel was used in the descriptive statistics for the corresponding census, births and personnel were determined.

	Census	Births	Personnel
Mean	144.095	874.045	861.500
Median	102.500	480.000	589.500
Standard Deviation	149.566	1063.666	821.597
Mode	28.000	0.000	328.000
Range	1104.000	5699.000	4037.000
Variance	22370.036	1131384.556	675021.618
Coefficient of Variance	1.038	1.217	0.954
Min	2.000	0.000	50.000
First Quartile	47.750	0.000	314.000
Third Quartile	181.750	1309.250	1095.250
Max	1106.000	5699.000	4087.000

For the data set census; Mean= 144.095, Median= 102.5, mode= 28, range= 1104, standard deviation= 149.566, variance= 22370.036 and Coefficient of Variance= 1.038. The five-number summary for census is thus {min=2, first quartile= 47.75, median= 102.5, third quartile= 181.75, max= 1106}.

For births; mean= 874.045, median= 480, mode= 0, range= 5699, standard deviation= 1063.666, variance= 1131384.556, coefficient of variance= 1.217. The five-number summary for births is thus {min= 0, first quartile= 0, median= 480, third quartile= 1309.25, max= 5699}.

For personnel; Mean = 861.5, median= 589.5, mode= 328, range= 4037, standard deviation= 821.597, variance= 675021.618, coefficient of variance= 0.954. The five-number summary for personnel is thus {min= 50, first quartile= 314, median= 589.5, third quartile= 1095.25, max= 4087}. (Shi, & McLarty, January 01, 2009).

“Outliers in the data are identified by first determining the Inter-Quartile Range (IQR). This is given by the difference between the third quartile and the first quartile. The IQR is then used to define the lower bound and the upper bound. The lower bound is determined by subtracting 1.5 of the IQR from the first quartile. To find the upper bound, 1.5 of the IQR is added to the third quartile. Any value of the data that lies below the lower bound or above the upper bound is considered an outlier” (MacRae, Welford & MacRae, 2011).

From the census data, IQR= 181.75-47.75 = 134, 1.5×IQR=201

Lower bound = 47.75-201 = -153.25, Upper bound = 181.75+201 =382.75

The outliers in the census data are 461, 414, 460, 390, 418, 456, 1106, 516, 395, 923, 416, 797 and 523.

For the births data, IQR = 1309.25-0 =1309.25, 1.5×IQR= 1963.88

Lower bound = 0-1963.88 =-1963.88 and upper bound = 1309.25+1963.88 =3273.13

The outliers in the births data are 3810, 3966, 3714, 3968, 3655, 5699, 3346, 3311 and 4207.

For the personnel data, IQR = 1095.25-314= 781.25, 1.5×IQR =1171.88

Lower bound= 314-1171.88 = -857.88, upper bound= 1095.25+ 1171.88= 2267.13

The outliers in the personnel data are 2310, 3694, 3486, 3301, 3928, 2581, 2534, 2620, 3123, 2745, 4087, 3012, 3090, 2312 and 3516.

The data for census, births and personnel can be presented on scatter plots where the outliers can clearly be observed.

Part 3: Inferential Statistics

1) To construct a confidence interval for the estimation of the average census for hospitals, the student-t test is used since the population standard deviation is unknown. The confidence interval for average of census would thus be given by;

μ =X̄ ± t(s/√n) where x̄ is the sample mean, s is the sample standard deviation, n is the size of the sample and t is the p-value obtained from the student t-value tables with the stated confidence level and (n-1) degrees of freedom.

For the 90% confidence level, α=0.1 and degrees of freedom =199, t= 1.6525

Therefore, μ = 144.095 ± 1.6525(149.566/√200)

=144.095 ± 17.477

The confidence interval is [126.618, 161.572] (Shi & McLarty, 2009)

For the 95% confidence level, α=0.05 and degrees of freedom = (n-1) =199, t= 1.972

Therefore, μ = 144.095 ± 1.972(149.566/√200)

=144.095 ± 20.856

The confidence interval is [123.239, 164.951]. (Shi & McLarty, 2009)

It can be observed that the margin of error increases with increase in confidence level. This is because a high level of confidence has a high p-value, which is directly related to the error margin. For the 90% confidence level, the margin of error is 17.477 while for the 95% confidence level, the margin of error is 20.856. Hence, increase in confidence level increases the confidence interval. However, the point estimate of the mean remains unchanged. (Frederic, 2009).

2) The sample proportion of hospitals that are general medical is determined by dividing the number of general medical hospitals in the sample with the total number of hospitals in the sample. There are 168 general medical hospitals in the sample of 200 hospitals. This means that there is 168/200 = 0.84 . The confidence interval for the population proportion is given by the interval [p̂ ± Z√(p̂q̂/n)] where p̂ is the sample proportion, q̂= (1-p̂), Z is the z-score value corresponding to the given confidence level and n is the sample size. At 95% level of confidence, Z = 1.96, p̂= 0.84, q̂= 0.16 and n= 200.

The confidence interval for the population proportion is given by;

0.84 ± 1.96 √[(0.84×0.16)/200]

= 0.84 ± 0.05 = [0.79, 0.89]

The point estimate for the population proportion is given by the sample proportion. In this situation the point estimate = 0.84. The error in interval is given by the term Z√ (p̂q̂/n). The error in the above interval is 0.05.

3) In order to test the accuracy of the claim that the average hospital in the US averages more than 700 births per year, the first step would be to state the null and alternative hypotheses. The null hypothesis contains the (=) equal sign. Therefore, in this case the hypotheses are as follows;

H0: μ ≤ 700

H1: μ > 700 (claim)

Since the population variance is unknown, the test statistic is given by;

t = (x̄-μ0)/(s/√n) where x̄ is the sample mean, μ is the hypothesized value = 700, s is the sample standard deviation and n is the sample size. For the births, x̄ =Substituting for the values

t= (874.045-700)/ (1063.666/√200) = 174.045/ 75.125

t = 2.317

From the student t-tables, with α = 0.01 and (n-1) =199 degrees of freedom, t0, the 100(1-α) percentile is given by 2.3452.

We reject the null hypothesis if t ≥ t0. Since 2.317 < 2.3452, that is, t < t0, we fail to reject H0. The conclusion made is that there is not enough evidence to support the claim that the average hospital in the United States averages more than 700 births per year. (Lehmann, & Romano, 2010).

4) To test whether hospitals in the United States employ less than 900 personnel, start by formulating the hypotheses. The assumption that hospitals in the US employ less than 900 personnel represents the claim. Therefore, the hypotheses are as follows;

H0: μ ≥ 900

H1: μ < 900 (claim)

Since the population variance is unknown, it is appropriate to use the t-test. The test statistic is given by;

t = (x̄-μ0)/(s/√n). For the personnel, x̄ = 861.5, s = 821.597 and n = 200. μ0 is the hypothesized value. In this case, μ0 = 900. The test statistic is thus given by;

t = (861.5-900)/ (821.597/√200) = -38.5/ 58.096 t = -0.6627

From the student t-tables, with α = 0.10 and (n-1) =199 degrees of freedom, t0, the 100(1-α) percentile is given by 1.2858.

The null hypothesis is rejected if t ≤ -t0. Since -0.6627 > -1.2858, that is, t > -t0, fail to reject H0. The conclusion made is that there is not enough evidence to support the assumption that hospitals in the United States employ less than 900 personnel. (McRae, 2011).

Part 4: Conclusion

The maximum number of hospitals is within the Midwest area with sixty whereas the Southwest region has the smallest number of hospitals with three. A large number of the hospitals in the United States are non-governmental, not-for-profit at eighty six, with only a small number that are owned by the government with the number of eight teen. This data analysis indicates that it can be determined that 84% of the hospitals in the US are general medical hospitals while 16% are psychiatric hospital. We do not know the variance but if the variance of the population was known, it would help in making more confident assumptions about the situation as it pertains to the hospitals. Hypothesis tests indicate, it can be concluded that the average hospital in the United States averages less than 700 births per year. Hospitals in the United States employ 900 or more personnel.

Top of Form

Personnel 792 1762 2310 328 181 1077 742 131 1594 233 241 203 325 676 347 79 505 1543 755 959 325 954 1091 671 300 753 607 929 354 408 1251 386 144 2047 1343 1723 96 529 3694 1042 1071 1525 1983 670 1653 167 793 841 316 93 373 263 943 605 596 1165 568 507 479 136 1456 3486 885 243 1001 3301 337 1193 1161 322 185 205 1224 1704 815 712 156 1769 875 790 308 70 494 111 1618 244 525 472 94 297 847 234 401 3928 198 1231 545 663 820 2581 1298 126 2534 251 85 432 864 66 556 347 239 973 439 1849 102 262 885 549 611 330 1471 75 262 328 377 575 1916 2620 571 703 535 160 202 1330 370 3123 2745 815 576 502 808 50 728 4087 3012 68 3090 1358 576 284 145 2312 1124 336 415 1779 338 453 437 261 609 647 61 2074 2232 948 409 153 741 1625 538 789 395 956 362 144 229 396 2256 731 1477 102 106 939 392 3516 785 607 273 630 1379 1108 583 514 216 1593 1055 399 834 104

Hospital

number of personnel

Census 107 198 356 100 9 159 65 48 253 21 27 30 43 233 2 11 84 219 112 124 50 142 111 140 28 154 150 144 42 77 119 27 15 179 175 461 32 74 414 253 180 184 243 115 215 48 124 189 181 9 28 288 108 154 76 165 295 101 69 12 185 378 114 49 106 460 43 29 125 17 10 14 173 207 223 82 64 139 109 298 52 34 168 21 390 47 80 50 113 45 76 129 60 418 17 138 64 62 131 265 456 40 310 72 19 112 375 15 78 123 54 96 82 1106 30 56 36 127 180 59 127 37 13 100 47 194 172 516 120 179 140 78 68 186 91 340 254 108 61 174 306 28 395 923 335 46 316 416 74 86 38 147 232 138 38 245 171 51 28 797 56 69 40 163 231 523 31 43 66 231 11 144 43 185 82 49 24 63 274 93 86 28 25 181 39 302 80 63 31 170 203 296 83 84 29 187 77 104 85 47

hospital

number of patients

Births 312 1077 1027 355 168 3810 735 1 1733 257 169 430 0 2049 211 16 2648 2450 1465 0 1993 2275 1494 1313 451 1689 1583 2017 995 2045 1686 503 126 2026 1412 1517 0 0 2719 1074 1421 762 3194 496 1442 0 1107 2989 113 0 0 173 1064 759 1317 1751 0 0 714 99 2243 3966 1308 0 2514 3714 126 556 1327 415 216 339 1217 2641 790 520 35 1168 793 0 0 14 0 0 0 0 776 451 0 145 1284 1 319 2154 295 496 589 806 701 3968 0 0 3655 0 0 0 0 0 3063 169 66 827 570 0 0 0 342 494 0 0 0 0 286 235 339 398 1275 5699 1364 714 0 0 0 779 0 2202 3346 1071 352 254 0 0 699 2462 3311 0 4207 0 339 130 91 1143 0 0 509 1026 0 447 1161 0 922 562 78 0 2122 0 0 0 710 1165 466 1106 376 0 637 0 352 447 1227 963 3038 0 0 868 1189 2849 1728 2171 364 0 2993 0 1964 601 387 1946 545 0 838 51

Hospital

number of births per year