Peer 1 – Janice Harvey
confidence intervals
The sample size, n, is what determines which method is appropriate to use between z-scores and t-distribution. When the sample size is above 30 (n>30) and the standard deviation is known then a z-score can be determined. When the sample size is too small, less than 30, and the sample standard deviation is not known then a t-distribution would be needed to determine an accurate level of confidence. Per Pearson’s Business Statistics (2014) the sample size must be large enough to safely assume a normal distribution model, or risk a possibility of the data being skewed.
One example of an application of a confidence interval to production is in determining the probability of how many books in a shipment from a bindery will cut imprecisely. If a distribution company takes the previous shipment from the bindery of 1200 books as its sample it can use a z interval as the sample size is greater than 30. They will be able to determine from that sample the confidence that the pages of a certain percentage of books will be cut incorrectly and therefore defective.
References:
Sharpe, N. R., De Veaux, R. D., & Velleman, P. F. (2014). Business Statistics (3rd ed.). Upper Saddle River, NJ: Pearson Education, Inc
Peer 2 Luke Hovey
5-1 Discussion: Z or T Confidence Intervals?
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Overview
Statistical and quantitative methodologies focus on gathering data for statistical analysis. There are many different techniques used for measuring and analyzing data. The entire group of data collected and studied is a statistical term called population (Sharpe, De Veaux, & Velleman, 2014, p. 260). A sample is a part of this whole population group of data. Sample statistics help estimate population and lead businesses to make key decisions.
Decisions rely on making estimations about population. Statistical hypothesis testing determines whether there is enough evidence in a sample of data to infer that a certain condition is true for the entire population (Taeger & Kuhnt, 2014). Confidence interval indicates a range of values that’s likely to encompass this true value. A confidence interval tells us how optimistic we are in our estimate.
How are z and t confidence intervals different?
The z and t confidence intervals are both used in hypothesis testing. But, they are different in a few ways. A z confidence interval is the number of standard deviations from the mean data point (Andale, 2016). Using the z confidence interval requires that you know the standard deviation of the population. Also, the sample size should be above 30 in order for you to be able to use the z confidence interval. The t confidence interval provides us with a different sampling distribution for each small sample size (Montgomery, 2016). If the sample size is below 30 and the population standard deviation is unknown then statisticians use t confidence interval.
Choose one interval and give an example of how it could be applied within an operations or production environment that is different from those mentioned in the overview.
The t confidence interval applies to operations when the standard deviation is unknown. If the sample data size used in analysis is less than 30 then a t confidence interval. The formula for t confidence internal is x bar ± t (s/√n) (Sharpe et al., 2014). The critical value from a standard t distribution is t and s is a sample standard deviation.
For example, a newspaper publication analyzes their mobile app crashes per month from the previous year. The resulting data helps the firm make decisions on the future of this product. The sample size is 12 months. The company uses a 95% confidence interval of 173.98 to 355.19 mobile app crashes to determine if the digital product is meeting their quality standards for each month. The descriptive statistics for this example is below.
| Month | Mobile App Crashes | Mobile App Crashes | ||
| January | 624 | |||
| February | 374 | Mean | 264.58 | |
| March | 267 | Standard Error | 41.17 | |
| April | 230 | Median | 237.50 | |
| May | 257 | Mode | #N/A | |
| June | 160 | Standard Deviation | 142.60 | |
| July | 401 | Sample Variance | 20,335.36 | |
| August | 188 | Kurtosis | 2.86 | |
| September | 161 | Skewness | 1.61 | |
| October | 245 | Range | 508.00 | |
| November | 116 | Minimum | 116.00 | |
| December | 152 | Maximum | 624.00 | |
| Sum | 3,175.00 | |||
| Count | 12.00 | |||
| Lower bound | 173.98 | |||
| Upper bound | 355.19 | |||
| Confidence Level (95.0%) | 90.61 | |||
The mean for this statistical analysis is 264.58 mobile app crashes per month. The standard deviation for the sample data set is 142.60 mobile app crashes with a range of 173.98 to 355.19 users who experience the mobile app crash.
References
Andale, A. (2016). T-Score vs. Z-Score: What’s the Difference? Retrieved February 15, 2017, from http://www.statisticshowto.com/when-to-use-a-t-score-vs-z-score/
Montgomery, J. (2016). Confidence Intervals with the z and t-distributions. Retrieved February 15, 2017, from http://pages.wustl.edu/montgomery/articles/2757
Sharpe, N. R., De Veaux, R. D., & Velleman, P. F. (2014). Business Statistics (3rd ed.). Upper Saddle River, NJ: Pearson Education, Inc.
Taeger, D., & Kuhnt, S. (2014). Statistical hypothesis testing. Statistical Hypothesis Testing with SAS and R, 3-16.
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