Confidence Intervals and Hypothesis Testing - Statistics Project Example

Confidence Intervals & Hypothesis Testing Executive Summary This paper attempts to explore thevarious statistical measures using a data collected from a randomly selected sample of size 100. The subject of interest is to test that an average university student drinks 2.0 cups of coffee a day. In order to ensure the randomness of our chosen sample whole university student names were collected and then using random number a sample was drawn of size 100. The data collected through a group of friends who provided me the aid to conduct a survey. The survey result was then tabulated on the MS Excel 2007. Using Excel tools further analysis was conducted. It was a matter of my interest as I am a keen coffee drinker and can drink 4 cups in a day. So the overall process made me feel quite enthusiastic and motivated to do the work. In order to test the hypothesis, the significance level chosen was 0.05 while for constructing confidence interval a 95% confidence level was chosen. The data gathered through random sampling is attached in the Appendix. All the tests are conducted using Data Analysis ribbon on MS Excel. Since the sample is large enough, the hypothesis testing will employ ‘t” test due to the unknown standard deviation of population. The results showed that the sample chosen showed to be a true representative of the population. As the hypothesis testing procedure proved it that the null hypothesis is accepted and the population mean was found to be equal to 2 coffee cups per day. The statistical procedures help in the estimation of population estimates by using sample values. The sampling methodology also impacts the relationship between sample values and the associated population values. The random sampling procedure helps in avoiding any biasness that may arise in the model or else the researcher can bias the outcomes of the model by manipulating the sampling technique. Outputs The output is as follows. Descriptive Statistics The initial analysis conducted is the most basic and is known as ‘Descriptive Statistics”. Descriptive statistics are methods of summing up large sets of numerical (quantitative) information. No. of cups of coffee Mean 1.83 Standard Error 0.087392 Median 2 Mode 2 Standard Deviation 0.873921 Sample Variance 0.763737 Kurtosis 0.347495 Skewness 0.421622 Range 4 Minimum 0 Maximum 4 Sum 183 Count 100 Confidence Level(95.0%) 0.173405 Alpha 0.05 The descriptive statistics shows the information regarding the measures of central tendency, measure of dispersion, confidence interval and many more. Measure of Central Tendency Measures of central tendency Mean 1.83 Median 2 Mode 2 These measures show that the middle value, calculated as median is 2 cups per day. The most recurring value as represented by ‘mode’ is 2 and the average as calculated by mean is 1.83 or approximately 2 cups per day. Since the mean value and the median are approximately same, hence, both of them can be regarded as the best representative of the given sample. The sum of all values is found to be equal to 183 cups. Dispersion Measures The significance of the mean value to be the representative of the given sample can be assessed by using the dispersion measures. The range being the simplest measure of variation explains how distant the lowest value and the highest value in a data set are located. Variance and standard deviation, on the other hand, show how much each data point varies from the mean value calculated. Measures of variation Standard Deviation 0.873921 Variance 0.763737 Range 4 The standard deviation shows that each unit in the given sample varies by 0.873921 from the mean value. The sample variance is 0.763737 while the data range is 4 showing that the highest and the lowest value in the given sample differ by 4. However, the small value of standard deviation shows that the majority of the values lie near the mean value Histogram The following table was generated by MS Excel to prepare a histogram. Bin Frequency 0 4 0.444444 0 0.888889 0 1.333333 30 1.777778 3 2.222222 43 2.666667 4 3.111111 8 3.555556 3 More 4 The histogram shows a double peaked distribution of the given data values. Confidence Interval The rationale behind doing simple random sampling from a population and calculating sample statistics is to use that sample statistic for the prediction of population statistics. But the accuracy of such estimation is doubtful. However, constructing a confidence interval helps by providing a range of set values which are more likely to hold the parameter of population within consideration. These are constructed at a confidence level. For example, 95% confidence interval shows that if numerous samples are generated from the same population, then the sample statistics would be a approximately 95% a predictor of population parameter. The requirements for constructing confidence interval include selecting the confidence level and the sample statistic and calculating the margin of error. Confidence Level Alpha Margin of Error Lower Limit Upper Limit           95% 0.05 0.17340482 1.6565952 2.0034048 In the given data, the sample mean was selected as an estimator of population mean. At 95% confidence level, the population mean is expected to lie in the 1.83±0.1734 interval. So the estimated value of population mean lies within the range of 1.6565952 to 2.0034048. Hypothesis Testing x ̅ 1.83 s 0.873921 n 100 Significance level 0.05 t 1.66 µ 2 a) Hypotheses Null Hypothesis: The average number of cups of coffee drunk by a university student is equal to 2 Ho: µ=2 Alternate Hypothesis: The average number of cups of coffee drunk by a university student is not equal to 2. H1: µ≠2 b) Decision rule At, 0.05 significance level, conducting a two-tailed test for the significance of mean suggests that null hypothesis can be rejected if either the t statistics is greater than 1.66 or it is lesser than -1.66. c) Test Statistic Since the sample size is greater than 30, but the population standard deviation is not known so t statistics will be used for testing the hypothesis. Formula Computation First the standard error value is calculated and then it is used to calculate the test statistic, in the given case, it is “t” statistics. d) Conclusion Since the calculated value of test statistics is less than the critical value, hence we cannot reject the null hypothesis. So the criterion suggests accepting the null hypothesis. e) P-Value It is the minimal significance level at which the null hypothesis can be rejected. It equals the region to the left of t = -1.94526 in addition to the region to the right of t= 1.94526. Hence the p-value can be computed as Since the P value is greater than α. i.e. 0.9454 >0.05, we accept the null hypothesis i.e., the mean number of coffee cups drunk by population is equal to 2 per day (Triola, 356). Conclusion In order to apply different statistical testing procedures a random sample was drawn among the university students. The main object of study was the coffee drinking behavior of young students. Since the random sample provides true representation of the population from which it is drawn, hence the sample statistics can be used to estimate population estimates. The sample size is 100. The statistical measures show varying results for the given data. The measures of central tendency showed median and mean of the sample to be approximately equal. The measures of variability showed that the each sample data unit varied from the mean by 0.873921. The histogram predicted frequency distribution to be double peaked. The 95% confidence interval level showed that the population mean is expected to lie in the 1.83±0.1734 interval. The Histogram shows bimodal distribution of the sample values. The Hypothesis testing proved that the average number of coffee cups drunk by whole university students is equal to 2 per day. Statistical methods help in the analysis of data and in the estimation of unknown population measures by utilizing known sample measures. This paper also utilized these techniques and employed hypothesis testing, confidence interval, measures of central tendency, measures of dispersion, frequency distribution to sort out the different aspects of gathered sample data so that it may be employed to perform hypothesis testing for the estimation of population mean. Works Cited Triola, Mario F. Elementary Statistics Using the TI-83/84 Plus Calculator. 11th ed. Pearson Education, 2008. Print. Appendix 1: Sample Data Serial number No. of cups of coffee 1 2 2 1 3 0 4 3 5 2.5 6 1.5 7 1 8 2 9 2 10 3 11 4 12 3 13 1 14 1 15 2 16 2 17 2 18 1 19 1.5 20 2.5 21 3 22 2 23 1 24 1 25 2 26 3 27 1 28 2 29 1 30 3 31 1 32 2 33 3 34 1 35 2 36 2 37 2 38 2 39 2 40 2 41 2 42 2 43 2 44 2 45 2 46 2 47 2 48 2 49 1 50 1 51 1 52 1 53 1 54 1 55 0 56 1 57 0 58 1 59 1 60 1 61 1 62 1 63 1 64 1 65 1 66 1 67 2 68 2 69 2 70 2 71 2 72 2 73 2 74 2 75 2 76 2 77 2 78 2 79 2 80 2 81 2 82 2 83 2 84 2 85 4 86 0 87 4 88 1.5 89 1 90 2.5 91 3.5 92 1 93 2.5 94 3.5 95 4 96 2 97 3.5 98 1 99 2 100 3 Read More