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.
Colour	White	Black	Silver	Rose Gold
Frequency	100	132	86	32
Can the company conclude that their expected distribution was accurate using a 10% level of significance?
Chi Square Distribution Table
a. Calculate the test statistic.
 χ 2=
 
12.64
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

This problem is about performing a chi-square goodness-of-fit test to determine whether the observed frequencies match the company's expected distribution.

### Hypotheses

- **Null hypothesis \( H_0 \):** The company's anticipated distribution (30% white, 35% black, 20% silver, 15% rose gold) is correct.
- **Alternative hypothesis \( H_1 \):** The company's anticipated distribution is not correct.

### Step 1: Expected frequencies

The expected frequencies for each color can be calculated based on the expected percentages and the sample size \( n = 350 \).

- **White:** \( 30\% \times 350 = 0.30 \times 350 = 105 \)
- **Black:** \( 35\% \times 350 = 0.35 \times 350 = 122.5 \)
- **Silver:** \( 20\% \times 350 = 0.20 \times 350 = 70 \)
- **Rose Gold:** \( 15\% \times 350 = 0.15 \times 350 = 52.5 \)

| Colour    | Observed Frequency (O) | Expected Frequency (E) |
|-----------|-------------------------|------------------------|
| White     | 100                     | 105                    |
| Black     | 132                     | 122.5                  |
| Silver    | 86                      | 70                     |
| Rose Gold | 32                      | 52.5                   |

### Step 2: Chi-Square Test Statistic Formula

The chi-square test statistic is given by:

\[
\chi^2 = \sum \frac{(O - E)^2}{E}
\]

Where:
- \( O \) = observed frequency
- \( E \) = expected frequency

Let's compute the test statistic.

\[
\chi^2 = \frac{(100 - 105)^2}{105} + \frac{(132 - 122.5)^2}{122.5} + \frac{(86 - 70)^2}{70} + \frac{(32 - 52.5)^2}{52.5}
\]

\[
\chi^2 = \frac{(-5)^2}{105} + \frac{(9.5)^2}{122.5} + \frac{(16)^2}{70} + \frac{(-20.5)^2}{52.5}
\]

\[
\chi^2 = \frac{25}{105} + \frac{90.25}{122.5} + \frac{256}{70} + \frac{420.25}{52.5}
\]

Now, let's calculate the total sum.

\[
\chi^2 = 0.238 + 0.737 + 3.657 + 8.004 = 12.636
\]

So, the test statistic \( \chi^2 = 12.64 \) (rounded to two decimal places).

### Step 3: Determine the critical value

We need to find the critical value from the chi-square distribution table for a 10% level of significance (\( \alpha = 0.10 \)) and \( k - 1 = 4 - 1 = 3 \) degrees of freedom.

Using the chi-square table, the critical value for \( \alpha = 0.10 \) and 3 degrees of freedom is approximately **6.251**.

### Step 4: Conclusion

- **Test statistic \( \chi^2 = 12.64 \)**
- **Critical value = 6.251**

Since the test statistic \( 12.64 \) is greater than the critical value \( 6.251 \), we **reject the null hypothesis**.

### Conclusion:
The company **cannot** conclude that their expected distribution was accurate based on the data, as the observed preferences significantly differ from the anticipated distribution.

---

Do you want more details or have any questions?

### Relative questions:
1. What is the chi-square goodness-of-fit test used for?
2. How is the degrees of freedom calculated in a chi-square test?
3. What is the null hypothesis in this scenario?
4. How does increasing the sample size affect the chi-square test?
5. How does the significance level impact the chi-square test conclusion?

### Tip:
When performing a chi-square test, always ensure that expected frequencies are not too small (preferably greater than 5) to ensure the validity of the test.

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.
Colour: White-100, Black-132, Silver-86, Rose Gold-32
Can the company conclude that their expected distribution was accurate using a 10% level of significance?


Analyze whether the smartphone colour preferences of a random sample of 350 potential customers match the expected distribution using a chi-square goodness-of-fit test. Includes calculation of the chi-square statistic, critical value, and hypothesis testing with a 10% significance level.

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