The Homework3 for Statistics

发布于：2024-09-19 ⋅ 阅读:(188) ⋅ 点赞:(0)

P r o b l e m   1 : ‾ \underline{ {\mathrm{Problem~1:}}} Problem 1:​

The ability to read rapidly and simultaneously maintain a high level of comprehension is often a
determining factor in academic success of many high school students. A school district is
considering a supplemental reading program for incoming freshmen. Prior to implementing the program, the school runs a pilot program on a random sample of n = 100 n=100 n=100 students. The students were thoroughly tested to determine reading speed and reading comprehension. Based on a fixedlength standardized test reading passage, the following reading times (in minutes) and increases in comprehension scores (based on a 100-point scale) were recorded. (The data for this problem is loaded in Canvas under the file student.csv.) **Use this problem to answer question 1-3.

Student Speed Comprehension
1 13 78
2 15 69
3 15 68
4 12 81
5 13 77
6 13 103
7 7 84
8 9 67
9 8 76
10 14 87
11 14 88
12 7 87
13 11 84
14 15 79
15 9 78
16 14 80
17 8 81
18 8 72
19 6 75
20 6 88
21 10 93
22 5 81
23 11 88
24 5 89
25 9 83
26 13 74
27 8 75
28 8 73
29 15 78
30 11 104
31 7 83
32 11 81
33 8 102
34 5 85
35 8 68
36 7 71
37 9 93
38 8 87
39 6 78
40 6 85
41 8 95
42 7 94
43 12 71
44 11 83
45 9 89
46 10 97
47 12 73
48 15 93
49 5 67
50 13 92
51 14 74
52 9 70
53 14 59
54 8 98
55 11 89
56 7 73
57 9 73
58 11 85
59 13 94
60 8 61
61 9 80
62 13 65
63 13 68
64 10 74
65 7 79
66 11 75
67 8 91
68 10 100
69 6 86
70 12 79
71 8 82
72 11 94
73 8 64
74 10 78
75 15 82
76 12 89
77 5 70
78 14 74
79 13 78
80 7 89
81 5 91
82 12 77
83 8 64
84 14 84
85 14 79
86 13 75
87 9 76
88 12 86
89 13 88
90 12 75
91 10 82
92 15 63
93 15 95
94 11 82
95 9 95
96 11 94
97 15 82
98 5 71
99 11 96
100 7 90
  1. Construct the 95% confidence interval for the mean comprehension score for all incoming
    freshman in the district. What is the lower bound for this interval? (round answers to 4
    decimal places)

Answer to Question 1 ≤ μ ≤ 1\leq\mu\leq 1≤μ≤Answer to Question 2

  1. Construct the 95% confidence interval for the mean comprehension score for all incoming
    freshman in the district. What is the upper bound for this interval? (round answers to 4
    decimal places)

Answer to Question 2 ≤ μ ≤ 2\leq\mu\leq 2≤μ≤Answer to Question 1

  1. Plot the comprehension scores using a histogram and then answer the following True/False
    question. True or False: Based on the historgram on comprehension scores, the scores
    appear to be a random sample from a population having a normal distribution.

Answer to Question 1:

To construct the 95% confidence interval for the mean comprehension score, we follow these steps:

  1. Calculate the sample mean ( x ˉ \bar{x} xˉ):

    x ˉ = ∑ x i n = 8145 100 = 81.45 \bar{x} = \frac{\sum x_i}{n} = \frac{8145}{100} = 81.45 xˉ=n∑xi​​=1008145​=81.45

  2. Calculate the sample standard deviation ((s)):

    s = ∑ ( x i − x ˉ ) 2 n − 1 = 9923.15 99 ≈ 9.9617 s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}} = \sqrt{\frac{9923.15}{99}} \approx 9.9617 s=n−1∑(xi​−xˉ)2​ ​=999923.15​ ​≈9.9617

  3. Determine the critical t-value for a 95% confidence interval with ( d f = n − 1 = 99 ) (df = n - 1 = 99) (df=n−

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