Question

Studies show that massage therapy has a variety of health benefits and it is not too expensive. A sample of 14 typical one-hour massage therapy sessions showed an average cost of $61.50. Past data indicate that the population standard deviation for a typical one-hour massage therapy session is $7.15. Compute a 90 % interval estimate of the average cost of a typical one-hour massage therapy session. Show your steps and manual (hand and calculator) calculations in 4 decimal points. a. 2 b. Interpret your estimate in (a) above. C. What assumption, if any, is required for your estimate above to be valid? Why?

Answer 1

a) The formula for estimation is:

μ = M ± Z(s_M)

where:

M = sample mean
Z = Z statistic determined by confidence level
s_M = standard error = √(s²/n)

M = 61.5
Z = 1.65
s_M = √(7.15²/14) = 1.91

μ = M ± Z(s_M)
μ = 61.5 ± 1.65*1.91
μ = 61.5 ± 3.1515

90% CI [58.3485, 64.6515].

b) Interpretation:

we can be 90% confident that the population mean (μ) falls between 58.3485 and 64.6515.

c) Assumptions:

The sample are randomly selected

Since the population is normal hence , the sample is assumed as normal.

The Z table used for Z value calculatin is

Table of Standard Normal Probabilities for Negative Z-scores Table of Standard Normal Probabilities for Positive Z-scores acont 0 5040 0 5080 0 5120. 0 5160 34 0.0007 0.0003 0 0003 o c003 acoca 0.0003 00 5000 5199 0.5239 0.5279 0.5319 05350 0.000 3 3 0.0005 0.0005 0.0005 0.0004 .0004 0.0004 o0004 0.0004 D.DO03 0.1 n538 0.5438 0.5478 0.5517 0.5557 D.5596 05636 D.5675 0.5714 0.5753 06365 D6409 06443 g ocos o ncca 0 000: 06217 06793 0631 06480 30 0,0013 0.0013 0.0013 .0012 0,0012 0,0011 Q0011 0.0011 0 co10 D.0010 0.6554 06591 0,6628 0.6664 06700 0.6736 06777 D.6806 0.6844 0,6879 07427 a0r23 0 7257 g o023 o Do2s 0 004 o c021 0ף7ד 0.7324 07357 07180 0.7454 D 7486 -21 0.0035 0.0034 0.0033 0,0032 0.0031 0,003C 0,0029 00028 0.0027 0,0026 07 0.7580 0,7611 0,7642 0.7673 0,7704 07734 0.7764 0.7794 0.7823 0.7852 o 062 a nso o 8212 D3315 Ds340 o Doso a onss o cosz o rous q.0057 0.0054 0 00s1 n co49 ns159 0.8238 0.8264 0 8789 os350 -24 0.0082 0.0080 0,0078 0,0075 0.0073 0.0071 0,0069 0,0068 0.0066 0,0064 1.0 0.8413 0.8438 0.8461 0.8485 O.8508 0.8531 0.8554 0.8577 0.8599 0,8621 0.0139 0.0136 0.0132 0.0129 0.0125 00122 0.0116 0.0113 D.0110 0,8907 0 8925 08944 08062 0.8997 00119 n8849 08869 0.8888 D 8980 0.0179 .0174 0.0150 .014o 0.0143 U.9032 D.9131 0.9162 0.9177 U. 0.0158 0.90bb 0.9113 U.914 0.049 0.0287 0.0281 0.0274 0,0268 0.0262 0,0256 O 0250 0.0244 00239 D.0233 15 0.9332 0,9345 0,9357 0,9370 09382 0.9394 0,9441 0.9406 0.9418 0.9429 n554 o 0436 o 0415 0 0401 o0303. 9564 o059 0.0532 0.0526 0.0516 0.0495 00485 0.0475 0.0465 0,0455 0,9656 0.9678 0.9706 -16 0.0548 0.0505 1 S 0.9641 0,9649 0.9664 09671 0.9686 D.9693 0,9699 00735 00885 09797 09838 n ogrs ac761 2.0 0918 no821 o 9834 -1.2 0.1151 0.1131 0.1112 0.1093 0.1075 0.1056 0.1038 0.1020 0.1003 0.0985 2.2 0.9861 0,9864 0,9868 0.9871 0.9875 0.9878 09881 0.9884 0.9887 0,9890 0 0034 0.1562 01515 0 1401 o c022. 0 9022 1379 01587 01530 a.1492 0.1469 01446 0 1421 n o91s 09920 0 9924 09927 0 g020 0g93 1 D.1611 09 0.1841 0.1814 0.1788 d.1762 0.1736 0.1711 O.1683 0.1660 0.1635 2.5 0.9938 0.9940 0.9941 0.9943 09943 09946 0.9948 D.9949 0.9951 0.9952 7 0.2148 0.2420 0 2389 02358 0 2327 0.2296 0.2266 0.2236 0.2206 0.2177 n.9965 0.9966 o 0067 0.9068 0,9969 0.9970 0.9971 0.9972 0.9973 0.9974 0.2578 0.2514 0.2483 D.245 2.8 0.9974 d.9975 099976 0.9978 D.9979 D.9979 0.9980 0,998 0.2743 D.2709 d.254 0.2611 0.9977 .2d 0 3156 n9987 09087 D 9089 0.3121 04 0,3446 .3409 0.3372 0.3336 0.3300 0.3264 0.3228 0.3192 30 0.9987 0,9988 09988 0.9989 0.9989 0.9990 0,9990 . L 000 o004 0 9095 0.4202 0.4013 a0004 900. o g001 n og05 0 9g95 0.4602 0.4562 0.4522 a.4483 04443 04404 0.4364 0.4325 0.4286 D4247 0.9995 0,9996 09996 0.9996 09996 D.9996 0.9996 0,9907 0.4961 0.4761 D.4641 00 0.5000 0.4920 0.4880 04840 0.4801 Q4721 0.4681 0.9997 0,9997 0.9997 09997 09997 0.9997 D.9997 D.9997 0.9997 0,9998 Note that the probabilities given in this table represent the area to the LEFT of the z-score The area to the RIGHT of a z-score 1-the area to the LEFT of the z-score