Sample Size
As the sample size increases, thestandard error becomes smaller resulting in a narrower confidence interval. This leads to a more precise estimateof the population parameter.
Conversely, smaller sample sizes result in larger standard errors and consequently wider, and therefore less precise, confidence intervals.
This effect of sample size in Example 7.1 is illustrated in Table 7.3. For Example 7.1, the 95% confidence interval limits are illustrated for three different sample sizes (based on z-limits of ±1.96 and population standard deviation,σ = 21).
Table 7.3
Table 7.3 Effect of sample size on confidence interval precision (usingσ= 21 andz = ±1.96) Sample size (
Sample size (n)) SSttaannddaarrd d eerrrroorr CCoonnffiiddeenncce e lliimmiittss 50 __ √ __σn = 2.97 78 ± 1.96(2.970) 72.18≤ µ ≤ 83.82 300 __ √ __σn = 1.212 78 ± 1.96(1.212) 75.62≤ µ ≤ 80.38 500 __ √ __σn = 0.939 78 ± 1.96(0.939) 76.16≤ µ ≤ 79.84
The increase in sample size from 50 to 300 and finally to 500 grocery shoppers has significantly improved the precision of the confidence interval estimate for the true mean value of grocery purchases by narrowing the range which is likely to cover µ.
Standard Deviation Standard Deviation
If the population standard deviation (σ) issmall in relation to its mean, variability in the data is low. This will produce anarrower confidence interval and therefore amore precise estimate of the population mean, and vice versa.
Chapter 7 – Confidence Interval Estimation
Note: Whenever the population standard deviation (σ) is unknown, it is usually estimated by the sample standard deviation (s). The sample standard deviation has the same effect on the estimated standard error as the actual population standard deviation.
Example 7.2
Example 7.2 Car Commuter Car Commuter Time StudyTime Study
From a random sample of 100 Cape Town car commuters, the sample mean time to commute to work daily was found to be 35.8 minutes. Assume that the population standard deviation is 11 minutes and that commuting times are normally distributed.
(a) Set 95% confidence limits for the actual mean time taken by all car commuters in Cape Town to travel to work daily.
(b) Set 90% confidence limits for the actual mean time taken by all car commuters in Cape Town to travel to work daily.
Solution Solution
(a) Given _ x= 35.8 minutes,σ = 11 minutes andn = 100 commuters.
Find the standard error of the sample mean.
σ _ x= ___ σ
√ __
n
= _____ 11
√ _____ 100
= 1.1 min
From the z-table, the 95% confidence level equates toz-limits of ±1.96.
Thus the lower limit is: 35.8 – 1.96(1.1) = 35.8 – 2.156
= 33.64 minutes
and the upper limit is: 35.8 + 1.96(1.1) = 35.8 + 2.156
= 37.96 minutes.
Thus the 95% confidence interval is defined as 33.64 ≤µ ≤ 37.96 minutes.
Management Interpretation Management Interpretation
There is a 95% chance that a Cape Town car commuter takes, on average, between 33.64 and 37.96 minutes to travel to work daily.
Example 7.2 contd.
Example 7.2 contd.
(b) The measures of _ x= 35.8 minutes,σ = 11 minutes,n = 100 commuters and the standard error = 1.1 minutes are all unchanged. Only thez-limits will change.
From the z-table, the 90% confidence level equates toz-limits of ±1.645.
Thus the lower limit is: 35.8 – 1.645(1.1) = 35.8 – 1.81
= 33.99 minutes
and the upper limit is: 35.8 + 1.645(1.1) = 35.8 + 1.81
= 37.61 minutes.
Thus the 90% confidence interval is defined as 33.99 ≤µ ≤ 37.61 minutes.
Management Interpretation Management Interpretation
There is a 90% chance that a Cape Town car commuter takes, on average, between 33.99 and 37.61 minutes to travel to work daily.
Applied Business Statistics
Note that the confidence limits are narrower (i.e. more precise) in (b) (90% confidence) than in (a) (95% confidence). Logically, the more confident a decision maker wishes to be that the true mean is within the derived interval, the wider the limits must be set. The trade-off that must be made is between setting too high a confidence level and creating too wide an interval to be of any practical use.
Example 7.3 Coalminers’ Employment Period Study Example 7.3 Coalminers’ Employment Period Study
A human resources director at the Chamber of Mines wishes to estimate the true mean employment period of all coalminers. From a random sample of 144 coalminers’
records, the sample mean employment period was found to be 88.4 months. The population standard deviation is assumed to be 21.5 months and normally distributed.
Find the 95 % confidence interval estimate for the actual mean employment period (in months) for all miners employed in coal mines.
Solution Solution
Given _ x= 88.4 months,σ = 21.5 months andn = 144 miners.
Find thestandard error of the sample mean.
σx_= ___ σ
√ __
n = _____ 21.5
√ _____ 144
= 1.792 months
From thez-table, the 95% confidence level equates toz-limits of ±1.96.
Thus the lower limit is: 88.4 – 1.96(1.792) = 88.4 – 3.51
= 84.89 months
and the upper limit is: 88.4 + 1.96(1.792) = 88.4 + 3.51
= 91.91 months.
Thus the 95% confidence interval is defined as 84.89 ≤µ ≤ 91.91 months.
Management Interpretation Management Interpretation
There is a 95% chance that the average employment period of all coalminers lies between 84.89 and 91.91 months.
Example 7.4
Example 7.4 Radial TRadial Tyres yres TTread Life Studyread Life Study
A tyre manufacturer found that the sample mean tread life of 81 radial tyres tested was 52 345 km. The population standard deviation of radial tyre tread life is 4 016 km and is assumed to be normally distributed.
Estimate, with 99% confidence, the true mean tread life of all radial tyres manufactured.
Also interpret the results.
Solution Solution
Given _ x= 52 345 km,σ = 4 016 km andn = 81 radial tyres.
Find thestandard error of the sample mean.
σx_= ___ σ
√ __
n = _____ 4 016√ ___
81
= 446.22 km
From thez-table, the 99% confidence level equates toz-limits of ±2.58.
Chapter 7 – Confidence Interval Estimation
The tyre manufacturer can be 99% confident that the actual mean tread life of all their radial tyres manufactured is likely to lie between 51 193.75 and 53 496.25 km.
7.6
7.6 The Rationa The Rationale of a le of a Confidence Confidence Interval Interval
A common question is: ‘How can we be sure that the single confidence interval will cover the true population parameter at the specified level of confidence?’ The car commuter time study (Example 7.2) is used to answer this question through illustration.
We begin by assuming that the population mean time to commute to work daily by Cape Town motorists is known to be 32 minutes (i.e. µ = 32).
Now construct 95% confidence limits around this true mean, µ = 32. These 95 confidence limits become: 32 – 1.96(1.1) ≤µ ≤ 32 + 1.96(1.1)
∴ 32 – 2.156 ≤µ ≤ 32 + 2.156
∴ 29.84 ≤ µ ≤ 34.16 minutes
Next construct a set of 95% confidence intervals on the basis of:
_ x – 1.96(1.1) ≤µ ≤ _ x+ 1.96(1.1)
Use a random selection of sample mean commuting times where:
– some fall inside the confidence limits of 29.84 and 34.16 minutes – others fall outside the confidence limits of 29.84 and 34.16 minutes.
Table 7.4 illustrates a selection of sample mean commuting times and their respective 95%
confidence intervals.
Table 7.4
Table 7.4 Car commuter times – sample means and their 95% confidence intervals Sample means within the interval
(usingz-limits = ±1.96)-limits = ±1.96) 29.84
(usingz-limits = ±1.96)-limits = ±1.96) 28.4
35.5
26.24≤µ≤ 30.56 33.34≤µ≤ 37.66
Applied Business Statistics
From Table 7.4, the following will be observed:
– Each 95% confidence interval derived from a sample mean that falls within or at the limits of the interval 29.84 ≤µ ≤ 34.16 willinclude the actual mean commuting time of 32 minutes. The sample means that are equal to 29.84 minutes and 34.16 minutes also cover the population mean, but at their upper and lower limits respectively. Since there is a 95% chance that a single sample mean will fall within the limits of 29.84 ≤ µ ≤ 34.16 minutes as shown above, there is a 95% chance that a single confidence interval will cover the true population mean.
– Each 95% confidence interval derived from a sample mean that falls outside the interval 29.84 ≤µ ≤ 34.16 willnot cover the actual mean commuting time of 32 minutes. This is likely to happen for only 5% of sample means.
Figure 7.1 illustrates this with numerous intervals based on sample means for a given
95% lower limit 95% upper limit
A structed about any _ xfalling between thelowerand upper 95% confidence limitsaround the true population mean will
cover the population mean.
Confidence intervalsC constructed about any _ x
fallingabove the 95%
confidence upper limits willnotcover the true population mean.
Figure 7.1
Figure 7.1 Illustration of the confidence interval concept (using a 95% confidence level)
Chapter 7 – Confidence Interval Estimation
Thus a single confidence interval at, say, a 95% confidence level, can be interpreted as follows: ‘There is a 95% chance that the limits of thissingle confidence intervalwill coverthe true population mean.’
7.7
7.7 The The Student Student t -distribution -distribution
The confidence intervals constructed in section 7.5 assumed that the population standard deviation,σ, was known. This measure was needed to calculate the standard error of the sample mean. However, it is often the case that the population standard deviation is unknown and needs to be estimated from the sample standard deviation,s. Then the student student t-distribution-distribution (ort-distribution), instead of thez-distribution, is used to derive the limits for the confidence level.