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A leading pharmaceutical company in the disposable contact lenses market has always taken for granted that the sales of certain peripheral products such as contact lens solutions would automatically go with the established brands. The long-standing culture in the company has been that lens solutions would not make a significant difference in user experience. Recent market research surveys, however, suggest otherwise. To gain a better understanding of the effects of contact lens solutions on user experience, the company conducted a comparative study in which 63 contact lens users were randomly divided into three groups, each of which received one of three top selling lens solutions on the market, including one of the company's own. After using the assigned solution for two weeks, each participant was asked to rate the solution on the scale of 1 to 5 for satisfaction, with 5 being the highest level of satisfaction. The results of the study are summarized below: $$ \begin{array}{|l|l|l|l|} \hline \text { Statistics } & \text { Sol. } 1 & \text { Sol. } 2 & \text { Sol. } 3 \\ \hline \text { Sample Mean } & x-1=3.28 & x-2=3.96 & x-3=4.10 \\ \hline \text { Sample Variance } & s 21=0.15 & s 22=0.32 & s 23=0.36 \\ \hline \text { Sample Size } & n_{1}=18 & n_{2}=23 & n 3=22 \\ \hline \end{array} $$ The mean satisfaction level of the combined sample of all 63 participants was \(x-=3.81\). Test, using the ANOVA F-test at the 5\% level of significance, whether the data provide sufficient evidence to conclude that not all three average satisfaction levels are the same.

Step by step solution

01

State the Hypotheses

First, we need to state the null and alternative hypotheses for the ANOVA test. The null hypothesis (\(H_0\)) is that all group means are equal: \(\mu_1 = \mu_2 = \mu_3\). The alternative hypothesis (\(H_a\)) is that at least one group mean is different from the others.

02

Calculate the Total Sum of Squares (SST)

The Total Sum of Squares (SST) measures the total variability in the data: \[ \text{SST} = \sum \sum (X_{ij} - \bar{X})^2 \]Where \(\bar{X} = 3.81\), and the sum is over all groups and all individuals.

03

Calculate the Between-Group Sum of Squares (SSB)

The Between-Group Sum of Squares (SSB) measures the variability between the group means: \[ \text{SSB} = \sum n_i (\bar{X}_i - \bar{X})^2 \]Where \(\bar{X}_i\) is the sample mean for each group, and \(n_i\) is the sample size for each group.

04

Calculate the Within-Group Sum of Squares (SSW)

The Within-Group Sum of Squares (SSW) measures the variability within each group: \[ \text{SSW} = \sum (n_i - 1) s_i^2 \]Where \(s_i^2\) is the sample variance for each group. This calculation uses the given variances \(s_{21} = 0.15\), \(s_{22} = 0.32\), \(s_{23} = 0.36\).

05

Calculate the Degrees of Freedom

Calculate the degrees of freedom for both between-groups and within-groups:- The between-groups degrees of freedom: \(df_B = k - 1 = 3 - 1 = 2\).- The within-groups degrees of freedom: \(df_W = N - k = 63 - 3 = 60\).Here, \(k=3\) is the number of groups and \(N=63\) is the total number of subjects.

06

Calculate Mean Squares and F-Statistic

Calculate the mean squares for between-groups and within-groups:- Mean square between groups (MSB): \(\text{MSB} = \frac{SSB}{df_B}\).- Mean square within groups (MSW): \(\text{MSW} = \frac{SSW}{df_W}\).Compute the F-statistic:\[ F = \frac{\text{MSB}}{\text{MSW}} \]

07

Determine the Critical Value and Make a Decision

Using the F-distribution table or software, determine the critical value for \(\alpha = 0.05\), \(df_B = 2\), and \(df_W = 60\). If the computed F-statistic is greater than the critical value, reject the null hypothesis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Hypothesis Testing

Hypothesis testing is a method used to decide if there's enough evidence to support a particular belief about a population parameter. In our contact lens study, we use hypothesis testing to compare the effects of different lens solutions on user satisfaction.

To begin, we establish two competing hypotheses:

• The null hypothesis (\(H_0\)) states that there is no difference in the means of satisfaction levels among the three lens solutions. Mathematically, that's expressed as \(\mu_1 = \mu_2 = \mu_3\).
• The alternative hypothesis (\(H_a\)) suggests that at least one solution leads to a different mean satisfaction level, \(\mu_i eq \mu_j\) for some \(i\) and \(j\).

The aim is to test whether we have sufficient evidence to reject the null hypothesis in favor of the alternative. If our statistical test indicates that the difference in means is significant, we can conclude that not all solutions have the same effect on user satisfaction.

Sample Variance

Sample variance is a measure that describes how much individual data points within a sample deviate from the sample mean. It is important because it tells us about the variability or spread in our data. Calculating variance is a crucial step in conducting an ANOVA test as it helps to assess the distribution of responses.

Here's how sample variance plays into our study:

• Each group has its own sample variance, calculated from their ratings of satisfaction.
• The group variances are given in the study: \(s_{21} = 0.15\), \(s_{22} = 0.32\), and \(s_{23} = 0.36\).

Sample variance is used in ANOVA to calculate the within-group sum of squares (SSW), which measures the variability within each lens solution group. The within-group variation helps us understand whether differences in satisfaction are due to variations within the groups or differences between the groups.

Statistical Significance

Statistical significance is a term that tells us whether the result of our study is likely due to chance or if it reflects a genuine effect. It is determined by comparing the calculated F-statistic from our ANOVA test to a critical value from the F-distribution tables at a chosen level of significance, often denoted by \(\alpha\).

In this exercise:

• The chosen level of significance is 5%, noted as \(\alpha = 0.05\).
• We calculate the F-statistic, which measures the ratio of between-group variance to within-group variance.
• The critical F-value is determined based on the degrees of freedom for our sample: 2 for the numerator and 60 for the denominator.

If the calculated F-statistic exceeds the critical F-value, we reject the null hypothesis, concluding that the differences in satisfaction among the lens solutions are statistically significant. This means the variation in satisfaction isn't just random but likely attributed to the type of lens solution used.