Question

Analyzing the most recent data, Jennifer’s manager Dr. Steinberg found out that in 2019 the anti-cancer prescription drug claims submitted by Healthy Life clients contained amounts considerably greater than he had anticipated. Specifically, he asked Jennifer Nguyen to check whether patients with Oncotype DX scores over 30 claimed, on average, more than $1000 in 2019. Using the systematic sampling method Jennifer selected 75 patients from this category (see the Major Assignment Data file). Please help Jennifer to test the claim that the population mean annual amount of the anti-cancer prescription drug claims was over $1000 in 2019. Use Data Analysis t-Test: Two-Sample Assuming Unequal Variances and “fool” Excel approach. Use 5% significance level. Is it possible to reach the same conclusion at 1% significance level? As you know, you have to make sure that the distribution of data is symmetric and bell-shaped in order to use t-distribution. Otherwise you have to use nonparametric methods for data analysis. Please build a histogram for the data using bins $500, $1000, $1500, $2000, $2500.

Monthly Claims

$1,254.17
$589.25
$1,047.12
$1,145.00
$1,055.25
$595.50
$1,050.00
$1,010.00
$1,290.90
$2,025.00
$520.00
$550.00
$1,050.00
$1,150.25
$1,150.00
$1,054.17
$1,150.00
$1,050.25
$1,190.90
$550.00
$1,250.00
$1,150.25
$1,150.25
$1,050.25
$1,150.25
$570.00
$1,550.00
$1,090.90
$1,054.17
$1,090.90
$1,054.17
$150.00
$270.00
$380.00
$2,190.90
$2,054.17
$1,150.25
$650.00
$1,150.25
$2,550.00
$600.00
$550.00
$550.00
$550.00
$650.00
$1,650.25
$1,550.25
$350.00
$250.00
$1,590.90
$1,554.17
$590.90
$1,554.17
$1,550.00
$1,510.00
$1,550.00
$520.90
$1,524.17
$1,510.25
$1,550.25
$1,276.45
$1,150.25
$1,150.25
$1,050.00
$1,554.17
$1,510.00
$2,100.00
$2,150.00
$1,150.25
$1,150.25
$1,075.00
$1,175.00
$1,100.00
$1,058.50
$1,125.25

Answer 1

Objective: To determine whether patients with Oncotype DX scores over 30 claimed, on average, more than $1000 in 2019.Let $\mu$ denote the average monthly claim. Wee need to test:

$H_{0}:\mu\leq 1000$ Vs   $H_{a}:\mu >1000$

The appropriate statistical test to test the above hypothesis would be a One sample t test, where we compare the mean of the population to a hypothesized value. But before running this test, we must ensure that the data is approximately normally distributed. We may check this assumption by constructing a Histogram as follows:

TICIOSOILLALL File Home Insert Page Layout Formulas Data Review View ww CV 19 LEL A 21 AZ | ZA A Sort Show Detail Data Analysis Hide Detail Solver Clear Reapply Advanced From Access Filter Connections Properties Refresh All Go Edit Links Connections From Web From From Other Text Sources Get External Data Existing Connections Group Ungroup Subtotal Text to Remove Data Consolidate What If Columns Duplicates Validation Analysis Data Tools Sort & Filter Outline Analysis E5 f B C E - H J к L M N o P Q R S ? X OK Bins 500 1000 1500 2000 2500 3000 Cancel A 1 Monthly Claims 2 $1,254.17 3 $589.25 4. $1,047.12 5 $1,145.00 6 $1,055.25 7 $595.50 8 $1,050.00 9 $1,010.00 10 $1,290.90 11 $2,025.00 12 $520.00 13 $550.00 D F Data Analysis 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 Help

Al Sort Text Colun ? File Home Insert Page Layout Formulas Data Review View Connections Clear A1 ZA Properties Reapply From From From From Other Existing Refresh Filter Access Web Text Sources Connections All- Edit Links Advanced Get External Data Connections Sort & Filter B1 х Histogram A Input OK 1 Monthly Claims Bins Input Range: $A$1:$A$76 2 $1,254.17 500 Cancel Bin Range: $B$1:$B$7 3 $589.25 1000 Help 4 $1,047.12 1500 Labels 5 $1,145.00 2000 Output options 6 $1,055.25 2500 Output Range: $D$2 7 $595.50 3000 New Worksheet Ply: 8 $1,050.00 New Workbook 9 $1,010.00 10 $1,290.90 Pareto (sorted histogram) Cumulative Percentage 11 $2,025.00 Chart Output 12 $520.00 13 $550.00 14 $1,050.00 15 $1,150.25 16 $1,150.00 $1054 17 17.

C IVI Bins 500 Bins Frequency 500 5 1000 Histogram 1500 1000 14 40 2000 1500 36 2000 14 30 2500 3000 2500 5 20 Monthly Claims $1,254.17 $589.25 $1,047.12 $1,145.00 $1,055.25 $595.50 $1,050.00 $1,010.00 $1,290.90 $2,025.00 $520.00 $550.00 $1.050.00 3000 1 10 500 1000 1500 2000 2500 3000 Monthly claims (in $s)

We find that the distribution of data is approximately symmetric and bell-shaped and hence, we may go for the t test.

Using excel: Since excel does not facilitate one sample t test, we may create a second group of comparison, with average monthly claim $1000, by entering each value as $1000. Also, in the new column created, since the variance would be zero, we may go for an Independent sample t test with unequal variances as follows:

Z ZA From Access Filter From Web From From Other Text Sources Get External Data Existing Connections A Sort Properties Refresh All- GO Edit Links Connections Reapply Advanced Text Colur Sort & Filter D2 fo K ? x OK Cancel А B 1 Monthly Claims Avg. Claim 2 $1,254.17 $1,000 3 $589.25 $1,000 4 $1,047.12 $1,000 5 $1,145.00 $1,000 6 $1,055.25 $1,000 7 $595.50 $1,000 $1,050.00 $1,000 9 $1,010.00 $1,000 10 $1,290.90 $1,000 $2,025.00 $1,000 12 $520.00 $1,000 13 $550.00 $1,000 14 $1.050.00 $1.000 с D H Data Analysis Analysis Tools Histogram Moving Average Random Number Generation Rank and Percentile Regression Sampling t-Test: Paired Two Sample for Means t-Test: Two-Sample Assuming Equal Variances t-Test: Two-Sample Assuming Unequal Variances z-Test: Two Sample for Means Help 00 01 11

02 x OK Cancel t-Test: Two-Sample Assuming Unequal Variances Input Variable 1 Range: $A$1:$A$76 Variable 2 Range: $B$1:$B$76 Hypothesized Mean Difference: Labels Alpha: 0.05 Help A B 1 Monthly Claims Avg. Claim 2 $1,254.17 $1,000 3 $589.25 $1,000 4 $1,047.12 $1,000 5 $1,145.00 $1,000 6 $1,055.25 $1,000 7 $595.50 $1,000 $1,050.00 $1,000 9 $1,010.00 $1,000 10 $1,290.90 $1,000 11 $2,025.00 $1,000 12 $520.00 $1,000 13 $550.00 $1,000 14 $1,050.00 $1,000 15 $1,150.25 $1,000 0 8 SD$2 * Output options Output Range: New Worksheet Ply: New Workbook

We get the output:

с D E F G t-Test: Two-Sample Assuming Unequal Variances Monthly Claims Avg. Claim 1128.88 1000 235714.20 0 75 75 0 A B 1 Monthly Claims Avg. Claim 2 $1,254.17 $1,000 3 $589.25 $1,000 4 $1,047.12 $1,000 5 $1,145.00 $1,000 6 $1,055.25 $1,000 7 $595.50 $1,000 8 $1,050.00 $1,000 9 $1,010.00 $1,000 10 $1,290.90 $1,000 11 $2,025.00 $1,000 12 $520.00 $1,000 13 $550.00 $1,000 14 $1,050.00 $1,000 15 $1,150.25 $1,000 16 $1,150.00 $1,000 17 $1,054.17 $1,000 74 Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail P(T<=t) two-tail t Critical two-tail 2.30 0.012 1.666 0.024 1.993

The Independent sample t test with unequal variances resulted in the test statistic value t = 2.30, with p-value 0.012 < 0.05. Since, the p-value of the test is significant, we may reject H₀ at 5% level. We may conclude that the data does provide sufficient evidence to support the statement that patients with Oncotype DX scores over 30 claimed, on average, more than $1000 in 2019.

If we were to test the hypothesis at 1% instead of 5%, we would have found the test result to be insignificant (since, p-value = 0.012 > 0.01) and hence, we would not be that supportive of the claim at 1% level. As obtained in the output, we find that the test statistic t = 2.30 < 2.378 does not lie in the rejection region and hence, we fail to reject H₀ at 1% level.

с D E F G t-Test: Two-Sample Assuming Unequal Variances Monthly Claims Avg. Claim 1128.88 1000 235714.20 0 75 75 A B 1 Monthly Claims Avg. Claim 2 $1,254.17 $1,000 3 $589.25 $1,000 4 $1,047.12 $1,000 5 $1,145.00 $1,000 6 $1,055.25 $1,000 7 $595.50 $1,000 $1,050.00 $1,000 9 $1,010.00 $1,000 10 $1,290.90 $1,000 11 $2,025.00 $1,000 12 $520.00 $1,000 13 $550.00 $1,000 14 $1,050.00 $1,000 15 $1,150.25 $1,000 16 $1,150.00 $1,000 8 0 74 Mean Variance Observations Hypothesized Mean Difference df t Stat P(T<=t) one-tail t Critical one-tail PIT<=t) two-tail t Critical two-tail 2.299 0.012 2.378 0.024 2.644