A phone company plans on offering their new smartphone in four colours: white, black, silver and rose gold. They anticipate that 30% of shoppers will prefer white, 35% will prefer black, 20% will prefer silver and 15% will prefer rose gold. They perform market research by asking a random sample of 350 potential customers which colour they prefer.

ColourWhiteBlackSilverRose GoldFrequency

101

126

89

34

Can the company conclude that their expected distribution was accurate using a 0.5% level of significance?

Chi Square Distribution Table

**a.** Calculate the test statistic.

 χ 2= χ 2= 

Round to two decimal places if necessary

**b.** Using the Chi Square Distribution Table, determine the critical value(s) for the hypothesis test.

 +

**c.** Conclude whether to reject the null hypothesis or not based on the test statistic.

Reject

Fail to Reject

To determine if the observed frequencies match the expected distribution, we'll use a χ² goodness-of-fit test.

### **a. Calculate the test statistic (χ²)**

1. **Null Hypothesis (H₀):** The distribution of color preferences matches the company's expected distribution.
2. **Alternative Hypothesis (H₁):** The distribution of color preferences does not match the company's expected distribution.

**Expected Frequencies Calculation:**

The expected frequency for each color is calculated using the expected proportion multiplied by the total number of customers surveyed (350).

\[
\text{Expected Frequency} = \text{Total Sample Size} \times \text{Expected Proportion}
\]

- **White:** \( 350 \times 0.30 = 105 \)
- **Black:** \( 350 \times 0.35 = 122.5 \)
- **Silver:** \( 350 \times 0.20 = 70 \)
- **Rose Gold:** \( 350 \times 0.15 = 52.5 \)

**Observed Frequencies:**

- **White:** 101
- **Black:** 126
- **Silver:** 89
- **Rose Gold:** 34

We calculate the test statistic (χ²) using the formula:

\[
\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}
\]

where \( O_i \) is the observed frequency and \( E_i \) is the expected frequency.

| Colour     | Observed (O) | Expected (E) | (O - E)  | (O - E)²  | (O - E)² / E |
|------------|--------------|--------------|----------|-----------|--------------|
| White      | 101          | 105          | -4       | 16        | 0.15         |
| Black      | 126          | 122.5        | 3.5      | 12.25     | 0.10         |
| Silver     | 89           | 70           | 19       | 361       | 5.16         |
| Rose Gold  | 34           | 52.5         | -18.5    | 342.25    | 6.52         |
| **Total**  | **350**      | **350**      |          |           | **11.93**    |

Thus, the test statistic is \( \chi^2 = 11.93 \).

### **b. Determine the critical value(s)**

The degrees of freedom (df) are calculated as:

\[
\text{df} = \text{Number of categories} - 1 = 4 - 1 = 3
\]

Using the Chi Square Distribution Table at a 0.5% level of significance (α = 0.005) and 3 degrees of freedom, the critical value is \( \chi^2_{0.005, 3} = 12.838 \).

### **c. Conclusion**

To determine whether to reject the null hypothesis, we compare the test statistic to the critical value:

- **Test statistic (χ²) = 11.93**
- **Critical value = 12.838**

Since the test statistic (11.93) is less than the critical value (12.838), we **fail to reject the null hypothesis**. There is insufficient evidence to conclude that the actual color preference distribution deviates from the expected distribution.

---

**Do you have any questions or would you like further details?**

### Follow-up Questions

1. What are the implications of failing to reject the null hypothesis in this context?
2. How can the company use the results of this hypothesis test for their marketing strategy?
3. What are the expected proportions based on, and why might they differ from observed preferences?
4. How is the critical value affected by the degrees of freedom and significance level?
5. What are some potential sources of bias that could affect the survey results?
6. How would increasing the sample size affect the reliability of the test results?
7. What are the assumptions underlying the χ² goodness-of-fit test?
8. How could the company refine their expected distribution for future studies?

### Tip

When conducting market research and statistical tests, ensure the sample size is large enough to provide reliable results and that the sample is representative of the target population.

Explore how to conduct a Chi-square goodness-of-fit test to determine if observed color preferences match expected distributions. Calculate χ² test statistic, compare with critical values, and interpret results.

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