Question

Assignment Score: 398/1200 Resources Check Answer Question'8 of 12 Let's illustrate the idea of a sampling distribution in the case of a very small sample from a very small population. The population is the scores of 10 students on an exam, shown in the table. Student 2 3 4 5 6 7 8 10 Score 63 94 81 45 72 72 75 59 65 51 4 drawn from the The parameter of interest is the mean score u in this population. The sample is an SRS of size n population. The Simple Random Sample applet can be used to select simple random samples of four numbers between 1 and 10, corresponding to the students. (a) Use the applet to generate 25 samples of size 4, calculate & for each, and construct a histogram of the 25 values. You are means resulted in the histogram provided. constructing the sampling distribution of &. One repetition of 25 sample Publisher: W H Freeman Ouestion Source: Moore, The BasicPractice Of Statistics, te Frequency OAssignment Score: 398/1200 Resources Check Answer Question 8 of 12 100 00 75 70 60 65 00 Sample Means 100 (b) Based on the histogram, what would you estimate to be the chance of obtaining a simple random sample of four closely students with x 2 782 Choose the description that matches these results most 100 O The chance of obtaining x2 78 is very likely The chance of obtaining x 2 78 is about fifty-fifty. 100 The chance of obtaining x 2 78 is very unlikely Question Sorce Moore, The Basic Practice Of Statistics, t Publisher WH Freema Assignment Score: 398/1200 Resources Check Answer K Question 8 of 12 (b) Based on the histogram, what would you estimate to be the chance of obtaining a simple random sample of four students with x2 78? Choose the description that matches these results most closely The chance of obtaining x 78 is very likely The chance of obtaining x78 is about fifty-fifty The chance of obtaining x 78 is very unlikely. 78 is almost certain The chance of obtaining x (c) Suppose you learn that students 1,3,5, and 7 are honors students. Would you regard their mean score to be "statistically significant"? The mean for these four students is 80.5. Based on the results in part (a), we might conclude that the mean score for the four honors students was unusually high (that is, this mean is statistically significant). O The mean for these four students is 80.5. Based on the results in part (a), we might conclude that the mean score for the four honors students was not at all unexpected (that is, this mean is not The mean for these four students is 78. Based on the results in part (a), we statistically significant) might conclude that the mean score statistically significant) not at all unexpected (that is, this mean is not for the four honors students was O The mean for these four students is 78. Based on the results in part (a), we might conclude that the mean score for the four honors students was unusually high (that is, this mean is statistically significant). PablisherWH.Ereeman Moore, The Baic Pactice Of Statiatics, e Ouestion Surce

Answer 1

a)

Using the simple random sample applet, the 25 samples are,

Samples	1	2	3	4	Mean
1	75	72	94	59	75
2	72	63	81	65	70.25
3	94	65	59	45	65.75
4	72	81	75	59	71.75
5	45	51	63	72	57.75
6	63	51	72	45	57.75
7	65	51	75	59	62.5
8	72	72	63	63	67.5
9	94	81	72	75	80.5
10	94	63	65	63	71.25
11	94	63	72	51	70
12	94	63	75	72	76
13	63	81	45	72	65.25
14	63	51	75	63	63
15	45	65	75	63	62
16	63	81	72	63	69.75
17	94	45	72	72	70.75
18	94	63	72	63	73
19	45	65	51	63	56
20	81	72	51	63	66.75
21	94	63	45	65	66.75
22	45	72	75	63	63.75
23	63	45	51	63	55.5
24	94	81	51	63	72.25
25	94	45	51	63	63.25

The histogram is obtained in following steps,

Step 1: Sort the data value,

Max	80.5
Min	55.5
Classes	10
Class width	2.5

Step 2: Count the data values withing the class

Bin	Frequency
55.5	1
58	3
60.5	0
63	3
65.5	3
68	4
70.5	3
73	5
75.5	1
78	1
80.5	1

Step 3: Make a column plot.

Histogram 6 4 3 2 1 55.5 58 60.5 63 65.5 68 70.5 73 75.5 78 80.5 LO

b)

Using the histogram given, the mean and standard deviation values are obtained as follows,

mid pint, m	frequency, f	f*m	m-mean	(m-mean)^2	f*(m-mean)^2
57	1	57	-11.84	140.1856	140.1856
61	3	183	-7.84	61.4656	184.3968
63	1	63	-5.84	34.1056	34.1056
65	7	455	-3.84	14.7456	103.2192
97	1	97	28.16	792.9856	792.9856
69	3	207	0.16	0.0256	0.0768
71	3	213	2.16	4.6656	13.9968
73	2	146	4.16	17.3056	34.6112
75	4	300	6.16	37.9456	151.7824
Sum	25	1721			1455.36

1721 f x m 68.84 mean, X 25

Efx (m - mean)2 1455.36 7 Std. Dev, =7.7872 S = 1 25 1

The probability X>= 78 is obtained by calculating z score,

78 68.84 =5.88 7.7872/25

The probability for the z score is obtained using the standard normal distribution table,

P> 5.88)=1- P(z < 5.88) 0.0000

Correct option: Hence the chances of obtaining
78
is very unlikely

c)

Correct Answer: The mean for these four students is 78. Based on the results in part (a), we might conclude that the mean score for the four honors students was unusually high( that is, the mean is statistically significant)

Explanation:

Student	Score
1	94
3	81
5	72
7	65
mean	78

Since the p-value for the z statistic is 0.000 < 0.05 at 5% significant level hence statistically significant