Analyzing Customer Preferences for Bath Soap Fragrances: A Chi-squared Goodness-of-Fit Test

Regal Manufacturing CC, a soap manufacturing company established in 2009, specializes in producing a variety of soap products including beauty soap, laundry soap, liquid hand soap, hospitality soap, and detergents. To enhance their marketing strategies, the company’s Marketing Director is interested in understanding customer preferences for bath soap fragrances. This valuable information would aid in making informed recommendations about the most preferred fragrances to offer in the market. To address this inquiry, the company conducted a survey by providing a free bar of soap with four different fragrance options to a random sample of 200 adult shoppers.

The fragrance options presented to the participants were:

1. Rose
2. Blue Lavender
3. Sandalwood
4. Lemon

The number of participants who chose each fragrance option are as follows:

Rose: 66

Blue Lavender: 53

Sandalwood: 45

Lemon: 36

In order to assess whether the distribution of preferences for these fragrance options follows any expected pattern or if there are significant deviations, a Chi-squared Goodness-of-Fit Test is conducted. The significance level chosen for this analysis is 2%, indicating a relatively strict threshold for accepting or rejecting hypotheses.

The Chi-squared Goodness-of-Fit Test is a statistical method used to determine whether observed categorical data (in this case, fragrance preferences) significantly deviate from the expected distribution. In our case, the expected distribution is assumed to be equal preferences for all four fragrance options (25% each) under the null hypothesis.

Hypotheses

Null Hypothesis (H0): The distribution of fragrance preferences is equal across all four options (Rose, Blue Lavender, Sandalwood, Lemon).

Alternative Hypothesis (Ha): The distribution of fragrance preferences is not equal across the options.

Calculating the Chi-squared Statistic

Using the formula for the Chi-squared statistic:

χ² = Σ((O – E)² / E)

Where:

• O = Observed frequency
• E = Expected frequency

For each fragrance option:

• Rose: χ² = ((66 – 50)² / 50) = 2.56
• Blue Lavender: χ² = ((53 – 50)² / 50) = 0.18
• Sandalwood: χ² = ((45 – 50)² / 50) = 1.00
• Lemon: χ² = ((36 – 50)² / 50) = 5.76

Summing up the individual χ² values gives the overall Chi-squared statistic:

χ²_total = 2.56 + 0.18 + 1.00 + 5.76 = 9.50

Degrees of Freedom (df): The degrees of freedom for a Goodness-of-Fit Test is calculated as the number of categories minus 1:

df = 4 – 1 = 3

Critical Chi-squared Value

At a 2% significance level and 3 degrees of freedom, the critical Chi-squared value from a Chi-squared distribution table is approximately 11.345.

Conclusion

Comparing the calculated Chi-squared statistic (9.50) with the critical Chi-squared value (11.345), we find that the calculated value is less than the critical value. Therefore, we do not have enough evidence to reject the null hypothesis. This implies that, at the 2% significance level, there is no significant difference between the observed and expected fragrance preferences among the given options.

In practical terms, the results suggest that customers’ preferences for bath soap fragrances are not significantly different from an equal distribution across the four options provided (Rose, Blue Lavender, Sandalwood, Lemon). Thus, the company can consider offering these fragrances in the market without favoring one over the others based on this survey.

It is important to note that the results of this analysis are based on the data collected from the specific sample of 200 adult shoppers. For more robust and generalizable insights, further studies with larger sample sizes and diverse demographics could be conducted.

In conclusion, the Chi-squared Goodness-of-Fit Test has provided valuable insights into customer preferences for bath soap fragrances, aiding Regal Manufacturing CC’s Marketing Department in making informed decisions regarding product offerings in the market.