Question 1165095
The correct assumption you had to make for this confidence interval to be valid, given a small sample size ($n=25$), is:

**b) The underlying population must have an approximately normal distribution.**

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### 📝 **Explanation**

When constructing a confidence interval for the population mean ($\mu$) using a small sample size ($n < 30$) and the **population standard deviation ($\sigma$) is unknown** (which is typically the case when a confidence interval is calculated in real-world scenarios, forcing the use of the $t$-distribution), the following conditions must be met for the interval to be statistically valid:

1.  **Random Sample:** The sample must be a simple random sample.
2.  **Normality:** The underlying population distribution must be **approximately normal**. 

[Image of a bell curve showing the normal distribution]


If the sample size were large ($n \ge 30$), the **Central Limit Theorem (CLT)** would ensure that the sampling distribution of the sample mean ($\bar{x}$) is approximately normal, regardless of the shape of the population distribution. However, with $n=25$, the CLT's guarantee of normality for the sampling distribution is not reliable unless the population itself is approximately normal.

* **Why the other options are incorrect:**
    * **a) Biased sampling distribution:** This is incorrect. A valid confidence interval requires an *unbiased* sampling distribution for the sample mean.
    * **c) Population must have an approximate t distribution:** This is incorrect. The population has its own distribution (which we assume is normal). The **sampling distribution** uses the $t$ distribution when $\sigma$ is unknown and $n$ is small.
    * **d) Sampling distribution of the sample mean must have a normal distribution:** While the sampling distribution *should* be approximately normal, you can only assume this if the population is normal (for small $n$) or if $n$ is large (by CLT). Since $n=25$ is small, you must assume the prerequisite condition—**population normality** (option b)—for the sampling distribution to follow the normal (or $t$) distribution.