Question

The objective of this is to carry out a hypothesis test using the Welch Approximate t Procedure, to determine if there is evidence of a difference in the caffeine content between sugar and diet soda.

** I know longer type question so posted it 3-4 times to help :) **

A) Use the Data Analysis ToolPak to carry out the Welch Approximate t Procedure. Include the output from Excel as a figure :

B) Identify the value of the test statistic from the Excel output :

C) Identify the p-value using the Excel output, and use it to determine whether or not the null hypothesis is rejected, assuming a 5% level of significance :

D) Calculate a 95% confidence interval for M1 - M2. Clearly state the point estimate for M1 - M2, the confidence coefficient, and the final confidence interval. To help with your calculations, use the rounded degrees of freedom that are reported as part of the Excel output. :

E) Provide an appropriate interpretation of the confidence interval you created. Does this interval support the results of your hypothesis test? Explain why or why not.

F) State the assumptions that must be satisfied for the inference procedures you just carried out to be valid. Using the plots you created as well as information provided in the question, is there reason to believe these assumptions have been violated? Justify your response

G) State a final conclusion about the caffeine content in sugar versus diet soda, using the results of this analysis :

Sugar	Diet
28.43	29.24
29.38	29.32
29.93	30.83
30.56	31.18
30.64	31.78
32.04	31.78
32.3	31.86
32.56	31.87
32.61	32.05
32.73	32.08
32.98	32.12
33.04	32.24
33.23	32.79
33.24	32.83
33.41	32.97
33.44	32.98
33.46	33.11
33.55	33.11
33.57	33.44
33.79	33.57
34	33.69
34.01	33.85
34.09	34.02
34.15	34.27
34.26	34.54
34.26	34.83
34.27	34.87
34.3	35.03
34.36	35.39
34.38	35.53
34.46	35.54
34.59	35.54
34.61	35.64
34.85	35.69
34.88	36.1
35.01	36.1
35.08	36.51
35.34	36.85
35.4	36.92
35.59	37.19
35.66	37.49
35.68	37.6
35.74	38.06
35.81	38.27
35.88	38.32
35.91	38.56
36.14	38.57
36.19	39.07
36.22	39.09
36.31	39.13
36.32	39.34
36.32	39.39
36.43	39.59
36.54	39.71
36.62	39.82
36.63	39.85
36.79	39.87
36.8	40.07
36.81	40.1
37.03	40.11
37.1	40.25
37.21	41.06
37.44	41.07
37.49	41.27
37.82	41.27
37.85	41.5
37.94	42.05
37.97	42.55
38.43	43.8
38.47	43.93
38.6	44.21
38.72	44.35
38.75	44.39
38.81	44.5
38.82	44.6
38.88	44.71
38.9	44.76
39.01	45.18
39.01	45.19
39.06	45.66
39.13	45.77
39.3	45.91
39.32	46
39.5	46.21
39.66	46.37
39.8	46.55
39.93	46.65
40	46.73
40.04	46.85
40.06	46.85
40.35	46.99
40.37	47.26
40.38	47.26
40.45	47.47
40.47	47.49
40.9	47.52
41.05	47.77
41.19	47.79
41.24	47.87
41.36	47.91
41.4	47.95
41.92	48.06
42.09	48.11
42.16	48.25
42.18	48.27
42.28	48.35
42.48	48.47
42.86	48.7
43.25	48.97
43.52	48.98
43.78	49.12
43.86	49.31
44	49.37
45.16	49.64
45.17	49.65
45.96	49.71
46.42	50.74
47.32	50.82
47.7	51.2
48.12	52.54

Answer 1

A) Using excel, running a Welch Approximate t Procedure:

Let $\mu _{1},\mu _{2}$ denote the mean caffeine content in Sugar and Diet soda. We have to test:

$H_{0}:\mu _{1}=\mu _{2}$ Vs $H_{a}:\mu _{1} \neq \mu _{2}$

BUURT - MICTUSUIL Excel View Home Insert Page Layout Formulas Data Review Connections Properties From From From From Other Existing Refresh Access Web Text Sources Connections All - So Edit Links Get External Data Connections Data Analysis Clear Reapply 1 EE 2show Detai Hide Detail z Sort Filter Y2 Advanced Group Ungroup Subtotal Text to Remove Data Consolidate What If Columns Duplicates Validation Analysis Data Tools Sort & Filter Outline Analysis F4 C D E G H J K L. MN O P Q R S T ? Data Analysis X A Sugar 28.43 29.38 29.93 30.56 30.64 32.04 32.3 32.56 32.61 OK Cancel Analysis Tools Histogram Moving Average Random Number Generation Rank and Percentile Regression Sampling t-Test: Paired Two Samole for Means TE-Test: Two-Sample Assuming Equal Variances t-Test: Two-Sample Assuming Unequal Variances z-Test: Two Sample for Means Help в Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 32.79 32.83 32.97 32.98 33.11 33.11 33.44 33.57 32.73 On min 00 oo 32.98 33.04 33.23 33.24 33.41 33.44 33.46 33.55 33.57 33.79 so

D2 C D E G H I J K ? 3 t-Test: Two-Sample Assuming Unequal Variances Input Variable 1 Range: $A$1:$A$121 Variable 2 Range: $B$1:$B$121 OK Cancel 6 Help Α Sugar 28.43 29.38 29.93 30.56 30.64 32.04 32.3 32.56 32.61 32.73 32.98 33.04 33.23 33.24 33.41 Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 Hypothesized Mean Difference: Labels Alpha: 0.05 8 9 10 11 12 13 14 15 16 Output options O Output Range: O New Worksheet Ply: O New Workbook 32.79 32.83 32.97

We get the output:

fix 1 R23 А Sugar 28.43 29.38 29.93 30.56 30.64 t-Test: Two-Sample Assuming Unequal Variances 3 Mean Diet 41.025 39.080 120 10 Sugar Sugar 37.625 16.228 120 0 203 -5.008 0.000 1.652 0.000 1.972 Variance Observati Hypothesi df t Stat P(T<=t) on t Critical o P(T<=t) tw t Critical t в Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 32.79 32.83 32.97 32.98 33.11 33.11 33.44 33.57 33.69 11 12 13 14 32.3 32.56 32.61 32.73 32.98 33.04 33.23 33.24 33.41 33.44 33.46 33.55 33.57 33.79 34 15 16 17 18 19 20 21 22

B) From the output, the test statistic is obtained as -5.008.

C) The p-value of the test is obtained as 0.000. Comparing the wih p-value with the significance level 0.05; Since, p-value = 0.000< 0.05, we may reject H0 at 5% level. We may conclude that there is a significant difference in the caffeine content in Sugar and Diet soda.

(D) The 95% CI for difference in mean $\mu _{1}-\mu _{2}$ is given by the formula:

$\mu _{1}-\mu _{2}\mp Z_{1-\alpha /2}SE_{(\mu _{1}-\mu _{2})}$

where,

$SE_{(\mu _{1}-\mu _{2})}=\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}$

Substituting the values,

$37.625 - 41.025\mp 1.96\sqrt{\frac{16.228}{120}+\frac{39.08}{120}}$

=( -2.069, -4.730)

Since the 95% CI does not contain the null value ($\mu _{1} - \mu _{2}={\color{Red}0 }$), we may conclude that the interval is significant. We arrive at the same conclusion as from the t test: Reject H0. We may conclude that there is a significant difference in the caffeine content in Sugar and Diet soda.

(E) Assumptions of Welch t test:

- The data (observations of both the independent groups)  is approximately normally distributed - The data collected is random

Checking the assumptions: We may check the normality assumption using a Histogram:

BUURT - MICTUSUIL Excel Home Insert Page Layout Formulas Data R eview View Clear Lad connections A1 AZZ 21 Sort Filter 21 From From Access Web From From Other Text Sources Get External Data Existing Connections Properties Refresh All Go Edit Links Connections Reapply Advanced BBB Text to Remove Columns Duplicates Valio Sort & Filter Da D E Data Analysis C Bins X 1 OK Cancel 13 A Sugar 28.43 29.38 29.93 30.56 30.64 32.04 32.3 32.56 32.61 32.73 32.98 33.04 Help в Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 Analysis Tools Anova: Two-Factor With Replication Anova: Two-Factor Without Replication Correlation Covariance Descriptive Statistics Exponential Smoothing F-Test Two-Sample for Variances Fourier Analysis Histogram Moving Average 10 11 12 13

Foc D E F G J KI C Bins H Histogram Input Input Range: $A$1:$A$121 $C$1:$C$30 OK Cancel Bin Range: Help Labels $E$4 * 10 11 13 A Sugar 28.43 29.38 29.93 30.56 30.64 32.04 32.3 32.56 32.61 32.73 32.98 33.04 33.23 33.24 33.41 33.44 33.46 33.55 33.57 33 79 B Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 32.79 32.83 32.97 32.98 33.11 33.11 33.44 33 57 Output options O Output Range: New Worksheet Ply: New Workbook Pareto (sorted histogram) Cumulative Percentage Chart Output 14 15 19

Similarly, for diet:

Н L M N O P C Bins Bins Frequency inloN00 Sugar A Sugar 28.43 29.38 29.93 30.56 30.64 32.04 32.3 32.56 32.61 32.73 32.98 33.04 33.23 33.24 33.41 33.44 33.46 33.55 33.57 B Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 32.79 32.83 32.97 32.98 33.11 33.11 33.44 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 11 12 Bins 11 14 15 16 17 18 19 20 9 11 40

Since the bell shaped graph is just a crude measure of normality, we might want to ensure that the condition is definitely satisfied. For this we may go for the summary statistics. If the mean and median of the population are almost equal, we would have a stronger evidence.

f - C Bins D E F G H I J K L M N O P Q 1 Bins Frequency 3 # Diet 7 3 Chart 2 A B Sugar Diet 28.43 29.24 29.38 29.32 29.93 30.83 30.56 31.18 30.64 31.78 32.04 31.78 32.3 31.86 32.56 31.87 32.61 32.05 32.73 32.08 32.98 32.12 33.04 32.24 33.23 32.79 33.24 32.83 33.41 32.97 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 0 1 2 3 Bins 5 6

DUUNI-IVIILIUJUILLALCI Home Insert Page Layout Formulas Data Review View E connections 21 41 Y clear ? E E E ? X Data Analysis Existing Connections 21 Sor From From From From Other Access WebText Sources Get External Data IT Properties Refresh All Edit Links Connections What-if Group Ungroup nalysis OK Cancel fx Help Analysis Tools Anova: Single Factor Anova: Two-Factor With Replication Anova: Two-Factor Without Replication Correlation Covariance Descriptive Statistics Exponential Smoothing F-Test Two-Sample for Variances Fourier Analysis Histogram - C Bins D E F G 2 1 ор T20 A Sugar 28.43 29.38 29.93 30.56 Bins Frequency 5 6 7 30.64 Sugar 10 11 B Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 32.79 32.83 32.97 32.98 33.11 33.11 22 AA 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 Bins 32.04 32.3 32.56 32.61 32.73 32.98 33.04 33.23 33.24 33.41 33.44 33.46 33.55 29 57 13 14 15 16 18 19

Descriptive Statistics Vie Input Input Range: Home Insert Page Layout Formulas Data Review Connections Le Properties From From From From Other Existing Refresh Access Web Text Sources Connections All Edit Links Get External Data Connections OK Cancel $A$1:$B$121 Columns O Rows Grouped By: at so Help Vhat if nalysis ✓ Labels in first row fax $I$16 D E F C Bins 1 Output options O Output Range: New Worksheet Ply: New Workbook ✓ Summary statistics Confidence Level for Mean: Kth Largest: Kth Smallest: 95 % Bins Frequency mm 9 TILL T20 A Sugar 28.43 29.38 29.93 30.56 30.64 32.04 32.3 32.56 32.61 32.73 32.98 33.04 33.23 33.24 33.41 33.44 33.46 33.55 33.57 33.79 34 B Diet 29.24 29.32 30.83 31.18 31.78 31.78 31.86 31.87 32.05 32.08 32.12 32.24 32.79 32.83 32.97 32.98 33.11 33.11 33.44 33.57 33.69 ot 25 27 29 31 33 35 37 39 41 43 45 47 49 51 Bins 5 7 8 9 0 -1 -2

Sugar Diet Mean Standard Error Median Mode Standard Deviation Sample Variance Kurtosis Skewness Range Minimum Maximum Sum Count 37.63 0.37 37.07 34.26 4.03 16.23 -0.08 0.38 19.69 28.43 48.12 4515.02 120.00 41.02 0.57 40.18) 31.78 6.25 39.08 -1.32 -0.08 23.30 29.24 52.54 4922.99 120.00

We may conclude that the normality assumption is satisfied.

(G) We have run the appropriate test for equality of means and have also ensured that the assumptions of the test are satisfied. Thus, based on the analysis, we may conclude that: There is a significant difference in the caffeine content in Sugar and Diet soda.