The confidence interval estimates a range within which an unknown population parameter is likely to lie. The confidence interval is calculated at a specified level of confidence, which measures the probability that the population parameter will fall within that range (Camm et al. 2018). Calculation of the mean's and proportion's confidence intervals relies on the central limit theorem, which assumes that large samples are normally distributed. Using the central limit theorem, the sample mean, has a distribution of N ( ). A level of confidence is required to find the critical value where is the level of significance. The paper uses mean and proportion confidence interval to describe data.
The formula for finding a confidence interval for the mean is:
(Camm et al., 2018). W here , the sample mean, s is the standard deviation, and n is the sample size
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The lower limit is while the upper limit is
Table 1 : Point estimate, standard deviation, and sample size for Springdale Mall
|
SPRILIKE, X7 |
|
| Mean, |
3.8133 |
| Standard deviation, s |
1.0643 |
| Sample Variance, |
1.1327 |
| Sample size, n |
150 |
The results imply that we are 95% confident that Springdale mall's mean general attitude will fall in the range ( 3.6430, 3.9837). Consumer perception towards Springdale mall is neutral but more towards like.
Table 2 : Point estimate, standard deviation, and sample size for Downtown
|
DOWNLIKE, X8 |
|
| Mean |
3.4067 |
| Standard Deviation |
1.1239 |
| Sample Variance |
1.2630 |
| Count |
150 |
The results imply that we are 95% confident that the mean general attitude towards Downtown will fall in the range (3.2268, 3.5865). Consumer perception towards Downtown is neutral but more towards dislike.
Table 3 : Point estimate, standard deviation, and sample size for West Mall
|
WESTLIKE, X9 |
|
| Mean |
3.2867 |
| Standard Deviation |
1.1834 |
| Sample Variance |
1.4005 |
| Count |
150 |
The results imply that we are 95% confident that the mean general attitude towards West Mall would fall in the range ( 3.0973, 3.4761) . Consumer perception towards West Mall is therefore neutral but more towards dislike.
The confidence interval for proportion measures a range within which an attribute is likely to occur in a population. The confidence interval is calculated based on the sample proportion, which is calculated by taking the ratio of a defined attribute, x, to the sample size, n ( Zikmund et al. 2013). Mathematically the formula for finding proportion is:
Where x is the number of sample subjects with a specific attribute, and n is the sample size.
The formula for calculating proportion confidence interval is:
( Zikmund et al., 2013).
Where is the critical value at (1- level of confidence interval at which the interval is calculated.
The lower limit is and the upper limit is
Table 4 : Break down of proportions for gender
|
X26 |
|
| Female,2 |
86 |
| Male,1 |
64 |
| Sample size, n |
150 |
| p =x/n |
0.5533 |
= 0.4267
0.0792 = (0.3475, 0.5058)
The results imply that we are 95% confident that the population proportion of males would fall within the range (0.3475, 0.5058).
Table 5 : Breakdown of proportions for marital status
| Married, 1 |
67 |
| Single & 0ther categories, 2 |
83 |
| Sample size, n |
150 |
| p =x/n |
0.5533 |
The results imply that we are 95% confident that the population proportion of single and other categories would fall within the range (0.4738, 0.6329).
Researchers use samples to represent a large population. The amount of sample collected must, however, be large enough to allow for generalizability. Assuming a specific margin error and a level of confidence, the minimum sample required can be estimated using the formula:
Sample Size, n = ( Zikmund et al., 2013).
Table 6 : Standard deviation and sample variance for Springdale Mall
|
SPRILIKE |
|
| Standard Deviation |
1.0643 |
| Sample Variance |
1.1327 |
Sample Size, n =
Where z is the critical value of z at a 95% level of confidence and E is the expected margin error and is the sample standard deviation.
The minimum sample size required for Springdale Mall is 1741.
Table 7 : Standard deviation and sample variance for Downtown
|
DOWNLIKE |
|
| Standard Deviation |
1.1239 |
| Sample Variance |
1.2630 |
Sample Size, n =
Where z is the critical value of z at a 95% level of confidence, and E is the expected margin error.
= 1.96
E = 0.05
is the variance.
Sample size, n = = 1940.84 1941
The minimum sample size required to obtain a margin error of 0.005 at a 95% confidence interval for Downtown is 1941.
Table 8 : Standard deviation and sample variance for West Mall
|
WESTLIKE |
|
| Standard Deviation |
1.1834 |
| Sample Variance |
1.4005 |
Sample Size, n =
Where z is the critical value of z at a 95% level of confidence, and E is the expected margin error.
= 1.96
E = 0.05
is the variance.
Sample size, n = = 2152.05 2152
The minimum sample size required to obtain a margin error of 0.05 at a 95% confidence interval for West Mall is 2152.
In conclusion, 95% population mean confidence interval suggests that Springdale has attracted the best consumers' attitude while Downtown and West Mall are neutral but towards dislike. The upper limit for males' proportion is approximately 50%, indicating that females are higher than males. To obtain a 0.05 margin error for mean at a 95% confidence interval, we would require sample sizes of 1741,1941, and 2152 for Springdale, Downtown, and West Mall, respectively.
References
Camm, J. D., Cochran, J. J., Fry, M. J., Ohlmann, J. W., and Anderson, D. R., 2018. Essentials of business analytics. Cengage Learning.
Zikmund, W. G., Carr, J. C., & Griffin, M. (2013). Business Research Methods. Cengage Learning.
