The file audit_fees provides selected financial data for a cross-section of 468 firms engaged in 'residential care activities for the elderly and disabled' in England.

The table below lists the first 10 firms from the file, where each row represents an individual firm. Column Firm_ID assigns a unique identification number to each firm. Column Turnover shows the operating revenue (turnover) in '000. Column Employees shows the total number of employees. Subsidiaries shows the number of subsidiaries in the corporate group. Audit_Fee shows the audit fees for the firms in the sample in '000. Audit fees are paid by firms to accounting firms and auditors for the provision of professional services, including the auditing of their financial accounts. The financial data are reported for the last financial year (2021).

Your employer is interested to know the extent to which firm turnover, the total number of employees and the number of subsidiaries affect audit fees paid by the firms in the sector.

Use the file audit_fees to answer the following questions:

• a.Run a multiple linear regression of Audit_Fee on Turnover, Employees and Subsidiaries. Based on your results, how would you write the estimated regression equation for the expected value of audit fee?
• b.Predict the expected audit fee paid by firms with turnover of 2 million that have 50 employees and two subsidiaries.
• c.What measure(s) can be used to assess a regression's ability to make good predictions? Comment on the goodness of fit of the estimated regression model.
• d.Test the null hypothesis that firm turnover has no (linear) effect on audit fees.
• e.Test the null hypothesis that the total number of subsidiaries has no (linear) effect on audit fees.
• f.What conclusions can you draw about the factors that affect Audit_Fee? Interpret the slope coefficient(s) of the statistically significant factor(s).

Firm_ID	Turnover ('000)	Employees	Subsidiaries	Audit_Fee ('000)
1	389062.00	9269	0	107.00
2	340612.00	11683	15	94.00
3	321900.00	983	11	100.00
4	178253.00	3328	2	47.00
5	155876.00	4147	32	23.00
6	153812.00	5281	3	294.00
7	153560.19	5040	9	94.80
8	127055.00	1645	24	215.00
9	123958.00	6140	0	167.00
10	122004.00	1720	4	67.00
11	97224.60	3364	24	52.77
12	95457.00	3486	10	96.00
13	88083.00	363	0	29.00
14	82706.00	567	3	112.00
15	81262.63	2171	46	78.43
16	67199.00	2344	0	48.00
17	67112.22	1829	8	27.30
18	66409.24	1164	64	36.00
19	61709.29	1918	22	125.00
20	58828.01	2204	25	51.50
21	58635.98	1804	22	83.94
22	56790.15	1941	24	5.00
23	53662.25	1422	2	56.40
24	52223.00	1576	3	48.00
25	50409.21	1751	24	69.65
26	49923.11	1826	3	23.00
27	49026.00	1653	64	8.00
28	47330.44	608	0	15.00
29	46932.00	1312	4	31.00
30	46718.00	1993	2	10.00
31	46138.00	1628	6	119.00
32	44627.00	1868	18	100.00
33	43988.82	1599	7	40.50
34	43451.00	1476	0	58.00
35	43394.93	1488	2	25.89
36	41822.00	974	7	85.00
37	41284.74	1031	13	63.90
38	40383.00	839	32	26.00
39	40379.00	1110	0	48.00
40	39430.52	1170	26	27.43
41	37424.32	1320	2	5.16
42	37283.00	1486	2	32.00
43	35731.86	677	0	33.00
44	35151.02	1783	0	33.30
45	34287.77	1068	13	38.00
46	32983.45	1029	29	15.90
47	31992.00	1601	0	17.00
48	31813.16	1416	0	60.00
49	30800.91	1175	3	29.00
50	30297.26	960	0	18.20
51	30260.53	1081	10	18.25
52	30222.00	720	0	31.00
53	29675.17	12	9	27.05
54	29662.00	835	0	21.00
55	29662.00	835	0	24.00
56	29383.00	523	0	109.00
57	29088.43	933	8	73.20
58	28969.00	750	0	19.00
59	28827.00	475	0	55.00
60	28468.00	532	0	32.00
61	27571.42	989	0	32.00
62	27091.66	1096	25	25.75
63	25732.00	685	2	57.00
64	25529.20	711	0	18.54
65	24833.00	826	2	29.00
66	24332.00	758	2	34.00
67	23585.00	979	45	15.00
68	23545.88	897	26	6.58
69	23449.41	850	7	17.60
70	23113.00	991	0	14.66
71	22896.91	628	29	14.88
72	22767.99	529	31	15.60
73	22456.20	681	8	5.78
74	22046.00	515	2	36.00
75	22001.46	503	19	26.98
76	21614.66	639	5	26.00
77	21470.02	570	13	7.46
78	21274.00	986	19	27.00
79	20890.00	545	0	25.00
80	20572.60	1115	24	11.00
81	20032.14	506	8	7.80
82	19799.00	367	2	56.00
83	19648.00	6	3	37.00
84	19506.46	903	9	12.00
85	19296.97	454	0	14.68
86	19284.71	579	20	33.00
87	18782.85	593	15	23.75
88	18326.94	716	0	26.00
89	17910.46	641	7	8.50
90	17654.32	812	3	12.75
91	17217.00	385	0	13.00
92	16587.00	322	5	26.00
93	15795.00	487	4	14.00
94	15764.18	308	0	5.25
95	15664.84	215	0	16.05
96	15452.10	559	755	19.05
97	15020.90	557	22	33.78
98	14428.72	430	8	20.53
99	13711.00	243	12	14.70
100	13632.82	462	29	17.98
101	13328.28	505	3	12.69
102	13304.90	476	0	25.00
103	13260.90	463	3	5.00
104	13178.00	719	45	8.00
105	13136.06	594	12	30.00
106	13045.43	322	7	18.60
107	13026.00	362	2	26.00
108	12895.22	444	3	8.92
109	12687.35	223	2	5.75
110	12687.35	223	2	8.50
111	12610.97	411	4	17.50
112	12471.82	401	9	16.92
113	12446.02	346	4	10.75
114	12446.02	346	4	21.25
115	12421.00	153	4	41.00
116	12067.90	381	7	18.62
117	12058.52	468	10	38.87
118	12056.63	410	3	17.10
119	11981.98	367	7	9.00
120	11963.79	294	0	14.24
121	11958.00	290	0	12.00
122	11895.47	367	1300	16.80
123	11709.54	391	22	24.69
124	11699.39	209	0	17.00
125	11654.00	225	6	2.00
126	11544.83	233	0	13.79
127	11444.94	328	11	12.03
128	11371.51	335	0	12.60
129	11360.77	408	3	12.90
130	11338.10	299	8	5.20
131	11061.70	319	0	19.50
132	11029.65	232	2	49.58
133	11010.80	375	8	37.00
134	10798.69	127	12	21.10
135	10639.13	340	0	34.25
136	10365.22	318	30	15.00
137	10348.31	513	0	17.39
138	10348.31	513	0	24.85
139	10342.90	342	0	21.91
140	10336.05	305	3	10.61
141	10143.98	282	0	10.59
142	10094.37	308	8	7.89
143	10061.98	332	4	12.30
144	10060.50	77	0	10.00
145	10025.00	38	0	13.00
146	9878.60	248	8	3.90
147	9847.98	227	7	15.50
148	9464.35	317	12	15.50
149	9179.28	304	9	9.90
150	9136.00	215	0	11.00
151	9063.55	296	51	11.82
152	8839.53	336	4	14.13
153	8836.88	337	3	9.85
154	8692.68	173	3	41.13
155	8556.49	278	12	17.50
156	8507.01	77	3	31.51
157	8498.45	249	8	4.60
158	8469.49	298	5	13.23
159	8430.71	71	425	16.80
160	8387.71	74	0	8.50
161	8285.50	322	10	15.40
162	8233.64	207	8	35.00
163	8045.95	208	4	12.20
164	7872.64	292	20	30.00
165	7840.00	176	17	87.00
166	7837.51	183	10	28.12
167	7743.33	160	46	2.70
168	7699.69	337	2	12.68
169	7691.69	204	7	11.07
170	7662.85	198	0	18.90
171	7618.42	236	3	43.26
172	7584.53	192	8	15.60
173	7491.52	326	3	4.95
174	7491.52	326	3	6.15
175	7435.00	236	0	15.00
176	7431.92	3	1459	6.94
177	7413.72	235	7	6.00
178	7324.94	209	98	22.98
179	7293.63	185	8	3.90
180	7213.98	245	7	14.32
181	6974.99	170	8	12.00
182	6944.00	207	2	25.00
183	6836.85	211	46	2.70
184	6834.75	308	2	13.70
185	6707.84	292	0	11.35
186	6673.02	125	3	16.50
187	6541.41	169	0	6.90
188	6541.41	169	0	7.90
189	6535.14	189	0	10.46
190	6488.96	226	6	13.74
191	6471.83	196	2	15.50
192	6381.99	161	0	6.40
193	6346.67	176	0	8.40
194	6273.00	35	0	7.00
195	6197.56	291	3	7.20
196	6146.38	166	7	9.60
197	6146.00	158	0	10.00
198	6125.04	174	2	9.26
199	6107.94	147	8	18.04
200	6098.51	321	0	10.11
201	6083.40	164	46	2.70
202	6071.48	156	3	16.38
203	6040.00	276	0	7.00
204	6013.83	212	13	6.10
205	5943.00	213	10	26.46
206	5818.78	219	8	6.79
207	5776.53	166	3	7.20
208	5760.70	160	46	2.70
209	5730.24	188	4	8.10
210	5700.54	137	0	5.74
211	5566.63	177	2	8.00
212	5513.40	102	4	6.91
213	5474.48	194	3	10.00
214	5451.73	190	0	18.04
215	5446.15	201	6	7.00
216	5441.81	175	4	43.20
217	5439.52	184	2	10.79
218	5409.60	127	2	6.00
219	5396.98	246	1300	14.00
220	5387.65	150	0	11.40
221	5337.89	182	0	9.45
222	5332.09	164	0	7.20
223	5306.45	124	3	8.50
224	5304.33	112	46	2.70
225	5189.42	139	0	6.00
226	5170.24	181	0	11.70
227	5141.78	119	0	16.80
228	5138.57	136	2	6.08
229	5138.57	136	0	8.08
230	5132.12	61	0	17.89
231	5103.39	126	0	5.40
232	5094.82	32	0	9.60
233	5037.31	74	840	6.00
234	5029.49	164	0	8.22
235	5023.34	173	0	10.68
236	5023.04	182	4	6.50
237	5023.04	180	4	14.28
238	5013.00	17	0	14.00
239	4966.14	108	46	2.70
240	4933.67	134	3	7.00
241	4924.25	121	100	17.80
242	4917.01	153	0	5.00
243	4885.71	160	12	17.51
244	4868.42	107	7	8.50
245	4829.78	106	0	10.50
246	4800.48	229	10	15.40
247	4779.63	125	8	28.70
248	4743.00	128	417	10.00
249	4736.78	104	3	24.71
250	4727.62	132	46	2.70
251	4723.73	18	0	17.00
252	4672.98	134	22	14.40
253	4644.25	165	0	9.99
254	4640.53	90	19	4.70
255	4624.12	146	7	7.22
256	4596.31	190	5	2.50
257	4582.09	3	1459	14.04
258	4552.66	91	0	3.77
259	4545.39	108	3	8.80
260	4511.74	141	6	3.60
261	4479.07	96	3	6.00
262	4469.40	26	3	15.11
263	4452.08	142	0	9.90
264	4446.42	162	8	9.50
265	4443.74	142	0	8.00
266	4367.96	72	0	6.00
267	4292.72	55	2	13.34
268	4246.49	100	2	15.66
269	4181.70	190	0	7.80
270	4149.41	170	5	8.50
271	4128.54	163	7	33.17
272	4087.62	110	16	20.40
273	4086.32	66	19	4.00
274	4053.66	181	4	9.20
275	4050.49	100	20	5.75
276	3984.41	118	5	5.00
277	3961.87	95	5	12.00
278	3865.48	134	0	4.70
279	3858.46	132	4	13.55
280	3856.37	70	0	12.80
281	3853.50	151	2	7.32
282	3837.31	156	8	3.95
283	3809.79	142	0	7.48
284	3800.27	85	2	4.18
285	3791.77	105	11	1.20
286	3770.24	91	0	5.74
287	3757.68	101	2	10.94
288	3751.97	105	5	6.00
289	3740.42	89	19	0.70
290	3719.59	88	2	11.16
291	3695.29	99	2	6.50
292	3692.75	96	3	13.84
293	3687.30	157	8	7.16
294	3685.07	107	0	8.10
295	3658.63	78	7	19.50
296	3629.79	137	2	13.00
297	3583.93	105	4	2.30
298	3583.93	105	4	2.30
299	3573.91	86	4	18.97
300	3531.87	149	2	7.00
301	3522.57	6	19	5.50
302	3515.19	96	7	19.50
303	3458.28	157	2	10.17
304	3413.48	106	3	19.14
305	3405.59	109	3	14.05
306	3347.45	165	0	4.00
307	3328.75	145	0	7.33
308	3266.44	72	4	19.27
309	3234.69	127	2	14.19
310	3229.98	150	7	6.47
311	3219.52	115	2	30.00
312	3218.69	108	0	3.00
313	3133.09	92	0	8.59
314	3122.65	81	0	7.70
315	3118.00	70	0	6.00
316	3108.00	112	0	2.00
317	3106.94	57	0	9.00
318	3099.77	70	0	5.90
319	3058.60	105	4	5.19
320	3058.60	105	4	8.35
321	3056.50	78	20	5.70
322	3021.89	84	20	3.00
323	3006.38	78	7	8.50
324	3000.61	59	4	3.77
325	2984.97	114	0	14.76
326	2977.96	96	46	2.70
327	2970.60	78	2	7.00
328	2891.26	84	3	1.84
329	2891.26	84	3	4.80
330	2878.81	89	4	10.90
331	2822.13	110	4	3.25
332	2812.56	92	3	10.88
333	2793.94	121	3	3.50
334	2770.42	57	0	18.49
335	2770.01	61	0	5.76
336	2765.58	68	3	5.94
337	2763.11	83	0	5.20
338	2748.17	11	0	16.40
339	2747.88	57	20	5.66
340	2737.53	79	0	6.42
341	2715.90	72	2	14.06
342	2686.50	62	0	5.60
343	2680.07	58	19	6.50
344	2672.62	66	63	8.34
345	2667.20	61	20	6.52
346	2654.81	93	16	19.20
347	2639.95	106	5	5.20
348	2635.27	93	0	5.97
349	2629.04	81	0	10.17
350	2619.22	3	4	30.00
351	2617.30	78	111	10.00
352	2611.42	63	0	3.00
353	2609.47	261	0	0.82
354	2584.62	104	0	9.40
355	2574.16	112	0	6.00
356	2542.00	68	0	5.00
357	2533.05	76	3	7.33
358	2532.00	94	0	2.00
359	2531.96	117	0	3.36
360	2526.48	72	0	2.00
361	2524.64	106	0	8.80
362	2523.19	81	2	2.80
363	2516.02	66	19	6.50
364	2495.70	59	3	3.60
365	2494.71	75	63	7.08
366	2443.20	74	9	24.00
367	2442.75	94	5	2.50
368	2392.69	83	0	11.75
369	2386.65	87	0	2.68
370	2384.98	61	19	0.70
371	2375.03	70	0	6.37
372	2368.21	56	4	7.57
373	2351.91	68	20	5.75
374	2351.60	80	9	16.40
375	2342.60	79	0	5.05
376	2309.29	70	0	9.00
377	2306.48	39	2	7.96
378	2305.83	118	0	10.58
379	2275.87	65	0	7.86
380	2257.82	80	7	34.80
381	2224.55	71	5	8.48
382	2210.21	73	2	6.14
383	2202.04	78	0	7.02
384	2187.87	82	0	3.20
385	2164.69	61	4	1.78
386	2157.00	112	0	7.00
387	2127.54	20	0	9.60
388	2119.35	50	6	9.00
389	2103.23	88	8	3.58
390	2038.00	54	0	5.00
391	2017.27	57	0	9.00
392	1998.15	60	98	8.34
393	1994.35	56	6	10.80
394	1991.40	63	0	2.40
395	1975.65	48	0	10.22
396	1959.09	49	50	7.80
397	1956.36	88	5	5.20
398	1954.70	37	25	7.96
399	1934.06	52	19	0.90
400	1925.93	56	0	6.00
401	1918.00	54	23	6.00
402	1912.84	153	0	8.71
403	1912.13	72	0	9.11
404	1903.16	58	0	5.95
405	1898.33	46	0	13.85
406	1892.76	86	0	22.72
407	1879.58	61	0	13.49
408	1866.88	49	0	4.50
409	1865.76	62	0	5.00
410	1847.60	69	0	7.05
411	1844.00	87	0	3.00
412	1815.00	66	0	6.00
413	1808.68	58	0	11.59
414	1787.31	66	2	6.00
415	1752.81	51	46	2.70
416	1752.57	60	3	20.50
417	1734.90	17	21	8.04
418	1729.34	64	50	9.90
419	1722.17	90	2	7.60
420	1721.94	67	0	3.67
421	1694.42	47	3	8.04
422	1692.81	58	0	4.80
423	1678.82	68	0	6.50
424	1678.03	58	0	3.41
425	1623.00	88	0	5.00
426	1600.04	61	7	16.80
427	1584.07	69	8	3.58
428	1567.26	51	0	2.75
429	1563.99	53	0	6.75
430	1556.37	75	63	6.88
431	1531.78	71	0	3.00
432	1518.00	81	0	10.00
433	1514.59	52	2	7.00
434	1509.41	40	0	5.91
435	1508.14	32	4	3.12
436	1478.59	68	0	5.10
437	1476.06	61	0	10.76
438	1466.71	27	0	5.06
439	1443.81	40	0	4.20
440	1429.73	26	0	3.11
441	1403.00	51	417	7.00
442	1396.98	54	0	5.40
443	1390.72	47	7	14.40
444	1389.61	9	0	6.46
445	1389.48	61	0	7.44
446	1373.11	60	0	3.60
447	1323.18	43	5	3.00
448	1316.59	38	2	5.91
449	1316.59	152	2	8.39
450	1315.00	32	4	3.00
451	1310.10	50	0	10.44
452	1302.90	40	0	5.50
453	1280.29	50	0	4.50
454	1263.85	47	0	4.50
455	1261.35	26	0	3.42
456	1250.68	32	0	9.80
457	1249.88	52	0	9.12
458	1212.30	37	25	7.96
459	1209.01	39	0	6.04
460	1166.36	37	0	6.00
461	1163.25	33	0	6.12
462	1102.20	28	0	4.27
463	1077.50	12	12	8.20
464	1071.70	42	0	6.36
465	1048.88	28	0	6.20
466	1023.03	38	0	12.03
467	1012.02	36	0	7.37
468	1006.48	34	0	7.08