統計學題目~~~~~~跪求統計高手解題順便翻譯..20點

Question

1.
The following is a sample of "bag weights" (in pounds)
for an industrial bagging operation:
50.6 49.8 50.8 50.5 50.2 50.4
50.6 51.4 50.4 50.3 49.9 50.1
50.8 50.8 50.6 50.8 52.2 50.7
50.8 50.6 50.7 50.6 50.3 49.8
50.8 50.6 49.1 51.2 50.2 52.0
49.8 50.8 49.0 51.1 46.8 50.5

a. calculate the sample mean y.
b calculate the sample variance s^2.
c. calculate the sample standard deviation s.
d. construct a histogram for the data.
e. construct a stem-and-leaf display of the data.


2.
National motors has equipped the ZX-900with a new disk brake system.
We define the stopping distance for a ZX-900 to be the distance (in feet) required to bring the automobile to a complete stop from a speed of 35 mph under normal driving conditions using this new brake system.
In addition, we define u to be the mean stopping distance of 60 feet.
National motors would like to claim in a new advertising campaign that the ZX-900 achieves a shorter mean stopping distance.
　Suppose that national motors randomly selects a sample of n=81 ZX900s.
The company records the stopping distance of each of these automobiles and calculates the mean and standard deviation of the sample of n=81 stopping distances to be y=57.8 feet and s=6.02 feet
a.　calculate 90%,95%,98%,and 99% confidence intervals for u
b.　using the 95% confidence interval, can national motors be at least 95% confident 　that u is less than 60 feet? Explain
c.　using the 98%confidence interval, can national motors be at least 98% . confident that u is less than 60 feet? Explain


3.
The bad debt ratio for a financial institution is defined to be the dollar value of loans defaulted divided by the total dollar value of all loans made.
A random sample of the seven Ohio banks is selected.
The bad debt rations (written as percentages) for these banks are 7,4,6,7,5,4, and 9%
a.　The mean bad debt ratio for all federally insured banks is 3.5%. Federal banking officials claim that the mean bad debt ratio for Ohio banks is higher than the mean for all federally insured banks. Set up the null and alternative hypotheses that should be used to justify this claim statistically.
b.　Assuming that bad debt ratios for Ohio banks are normally distributed, use the sample results given above to test the hypotheses you set up in part (a) with ａ=0.01. Interpret the outcome of the test.

Answer 1

1.
下列是一個工業裝袋作業的袋重樣本 (單位為磅)：

50.6 49.8 50.8 50.5 50.2 50.4
50.6 51.4 50.4 50.3 49.9 50.1
50.8 50.8 50.6 50.8 52.2 50.7
50.8 50.6 50.7 50.6 50.3 49.8
50.8 50.6 49.1 51.2 50.2 52.0
49.8 50.8 49.0 51.1 46.8 50.5

a. 計算樣本平均數 y
b. 計算樣本變異數 s^2
c. 計算樣本標準差 s
d. 繪製一個資料的直條圖
e. 繪製一個資料的莖葉圖

2.
國家汽車在 ZX-900 上裝置了新行的碟煞系統。我們以正常駕駛下時速 35 英里煞車至完全停止的距離為 ZX-900 的停止距離 (單位為英尺)。同時，定義 u 為平均停止距離 60 英尺。國家汽車想要在新的廣告中宣稱 ZX-900 擁有更短的平均停止距離。

假設國家汽車隨機選出 81輛 ZX-900 做為樣本，紀錄每一輛車的停止距離，然後求得這 81 量車停止距離的平均值與標準差，分別是 y=57.8 英尺以及 s=6.02 英尺。

a. 計算 u 的 90%, 95%, 98% 以及 99% 信賴區間
b. 以 95% 信賴區間來說，國家汽車是否有 95% 的信心認為 u 小於 60 英尺，請說明
c. 以 98% 信賴區間來說，國家汽車是否有 98% 的信心認為 u 小於 60 英尺，請說明

3.
金融機構壞帳率的定義為無法回收的帳款金額除以總帳款金額。七家俄亥俄銀行被抽出做為隨機的樣本，這幾家銀行的的壞帳率分別是 7, 4, 6, 6, 5, 4 以及 9 個百分點

a. 所有聯邦保險銀行的平均壞帳率為 3.5%，聯邦銀行的官員宣稱俄亥俄銀行的壞帳率較所有聯邦保險銀行的平均值還來得高。試設定虛無假設與對立假設，以統計方式來糾正這項說法。

b. 假設俄亥俄銀行的壞帳率為常態分佈，試以上述的樣本數字在 α=0.01 之下進行你在 a. 中所設計的假設檢定。請說明檢定結果。