Business Statistics Topic 6

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1 Business Statistics Topic 6
Estimation

2 Business Statistics:Topic 6
Learning Objectives By the end of this topic you will be able to: Explain basic estimation processes Estimate population parameters such as mean and proportion Use confidence interval estimation techniques when s is known & unknown Choose the appropriate sample size for estimation purposes Business Statistics:Topic 6

3 Business Statistics:Topic 6
Estimation Business Statistics:Topic 6

4 The Estimation Process
Population I am 90% confident that µ is between 25 and 35 µ = ? Sample mean, is 30 Draw Sample Business Statistics:Topic 6

5 Business Statistics:Topic 6
Point Estimates Estimate population parameters Using sample statistics Mean µ Standard Deviation  Proportion P S pS These are single value estimates They do not tell us how close our estimate is to the actual unknown parameters Business Statistics:Topic 6

6 Business Statistics:Topic 6
Interval Estimates As the sample statistic varies from sample to sample, an interval based on the value of the sample statistics provides an estimate of the population parameter. A level of confidence is selected such as 90%, 95% or 99% This tells us how close the estimates are to the true population parameter Business Statistics:Topic 6

7 Business Statistics:Topic 6
Interval Estimation Business Statistics:Topic 6

8 Confidence Interval for µ ( known)
Assumptions Population standard deviation  is known Population follows a normal distribution (or) Sample size is large Business Statistics:Topic 6

9 Confidence Interval for µ ( known)
(Sample mean)  (z).(Standard Error) Range Interval The value of ‘z’ changes according to the confidence level (90%,95% or 99%) Business Statistics:Topic 6

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Confidence Level 100 (1-)% Where  is the proportion of the tails (upper & lower) outside the confidence interval Business Statistics:Topic 6

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90% Confidence Level  is 10% Each tail is 5% Z = 1.645 Business Statistics:Topic 6

12 90% Confidence Interval Estimate of 
90% of similarly constructed intervals contain  and 10% do not Business Statistics:Topic 6

13 Business Statistics:Topic 6
Example The management of a famous restaurant wants to estimate the average amount a customer spends for dinner. The management believes that the amount spent by all customers (population) follows a normal distribution with a standard deviation of $5. They selected a random sample of 36 customers and found the sample mean to be $35. Business Statistics:Topic 6

14 90% Confidence Interval Estimate of 
90% confident that the value of the population mean m of the amount spent lies in this interval. Business Statistics:Topic 6

15 Business Statistics:Topic 6
Interval Estimation Business Statistics:Topic 6

16 Confidence Interval for µ ( unknown)
Assumptions Population standard deviation  is unknown Population follows a normal distribution (or) Sample size is large Business Statistics:Topic 6

17 Business Statistics:Topic 6
由于在实际工作中，往往σ是未知的，常用s作为σ的估计值，为了与Z转换区别，称为t转换，统计量t 值的分布称为t分布。 学生t-分布（Student's t-distribution）经常应用在对呈正态分布的总体的均值进行估计。 Business Statistics:Topic 6

18 Business Statistics:Topic 6
以0为中心，左右对称的单峰分布； t分布是一簇曲线，其形态变化与n（确切地说与自由度ν）大小有关。自由度ν越小，t分布曲线越低平；自由度ν越大，t分布曲线越接近标准正态分布（u分布）曲线，如图. 对应于每一个自由度ν，就有一条t分布曲线，每条曲线都有其曲线下统计量t的分布规律 Business Statistics:Topic 6

19 Business Statistics:Topic 6
k=120(正态） K=20 K=5 Business Statistics:Topic 6

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t分布的均值与标准正态分布均值相同，为0，但方差为k/(k-2)。由此，在求t分布的方差时定义自由度必须大于2。 标准正态分布的方差等于1，因此，t分布方差总大于标准分布的方差，也就是说，t分布比正态分布略“胖”些。 Business Statistics:Topic 6

21 Business Statistics:Topic 6
当k增大时，t分布的方差接近于标准正态分布方差值1。 例如： 当k=10时，t分布的方差为10/8=1.25； 当k=30时，t分布的方差为30/28=1.07； 当k=100时，t分布的方差为100/98=1.02； 结论：随着自由度的逐渐增大，t分布近似于正态分布。 注意：对于t分布，不要求其样本容量很大，k=30时，t分布与正态分布已很近似。 Business Statistics:Topic 6

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例：自由度为10，P(t>1.812)=P(t<-1.812)=0.05 P(︱t︱>1.812)=P(t>1.812)+P(t<-1.812)=0.1 -1.812 1.812 0.05 Business Statistics:Topic 6

23 Business Statistics:Topic 6
例：变量X表示面包房每日出售的面包量，在15天内，出售面包的样本方差为16。假定真实的出售量为70条，求任意15天内出售面包平均数量为74条的概率。 分析：本例中已知样本方差S²=16，则S=4，总体均值（真实的出售量）=70，运用t变量公式得： 查t分布表，自由度为(n-1)=15-1=14 当自由度为14时，查表得，t值大于等于2.977的概率为0.005，大于等于4.140的概率为0.0005，所以，t值大于等于3.873的概率介于0.0005~0.005之间。 Business Statistics:Topic 6

24 Confidence Interval Estimate of µ ( unknown)
Use ‘t’ instead of ‘z’ Use ‘s’ an estimate of  Sample mean  (tn-1).(Standard Error) Range Interval Business Statistics:Topic 6

25 Business Statistics:Topic 6
‘t’ Table 10/2 = 5%= .05 (lower left hand tail) 45% = 0.45 (upper right hand tail) 90% - t= t=2.015 Degrees of freedom, df = n-1 Match ‘df’ and upper tail area, a /2 For example, ‘t’ value with 90% confidence level & sample size, n = 6 a = 10% a /2 = 5% =.05 df=n-1=6-1=5 Match (5, .05) ‘t’ value = Business Statistics:Topic 6

26 Confidence Interval ( unknown)
Population follows a normal distribution Given 90% confidence level Sample size 6 Business Statistics:Topic 6

27 Business Statistics:Topic 6
Example The management of a famous restaurant wants to estimate the average amount a customer spends for dinner. The management believes that the amount spent by all customers (population) follows a normal distribution. They selected a random sample of 36 customers and found the sample mean to be $35 and sample standard deviation to be $5. Note that  is unknown: Use ‘t’ Business Statistics:Topic 6

28 90% Confidence Interval estimate of 
90% confident that the value of the population mean m of the amount spent lies in this interval. Business Statistics:Topic 6

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What do you notice? The confidence interval using ‘t’ is wider than the confidence interval using ‘z’ As ‘n’ increases the ‘t’ values moves closer to the ‘z’ value Business Statistics:Topic 6

30 Determining the sample size
With the previous example, if the management chooses a 99% confidence level, and with  $2 of sampling error, then they need to select a sample of at least size 42 e is sampling error Business Statistics:Topic 6

31 Interval Estimation of Population Proportion (p)
Business Statistics:Topic 6

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Example In an election poll a random sample of 500 people showed that 42 preferred voting for a particular candidate. Set up a 90% confidence interval estimate for the population proportion, p Business Statistics:Topic 6

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Example Business Statistics:Topic 6

34 Determining the sample size
Based on the previous example, what sample size is needed to be within ± 2% with 95% confidence? Business Statistics:Topic 6

35 Business Statistics:Topic 6
Summary In this topic you have discussed: Estimation processes Point estimates of population parameters Confidence interval estimate of  when  is known Confidence interval estimate of  when  is unknown Determining the sample size Confidence interval estimation of the population proportion Business Statistics:Topic 6