document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1165095>Question 1165095</A>: <I><SPAN STYLE=\"word-break: break-all;\"> . You've calculated a confidence interval for μ and obtained the range (1000, 2100). If your calculation was based on a sample of size of 25, what assumptions did you have to make for this interval to be valid? a) The sampling distribution must be biased with 24 degrees of freedom. b) The underlying population must have an approximately normal distribution. c) The population must have an approximate t distribution. d) The sampling distribution of the sample mean must have a normal distribution. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #853117 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=853117\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> The correct assumption you had to make for this confidence interval to be valid, given a small sample size ($n=25$), is:\r<BR>\n" );
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document.write( "**b) The underlying population must have an approximately normal distribution.**\r<BR>\n" );
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document.write( "### 📝 **Explanation**\r<BR>\n" );
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document.write( "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:\r<BR>\n" );
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document.write( "1.  **Random Sample:** The sample must be a simple random sample.<BR>\n" );
document.write( "2.  **Normality:** The underlying population distribution must be **approximately normal**. \r<BR>\n" );
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document.write( "[Image of a bell curve showing the normal distribution]\r<BR>\n" );
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document.write( "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.\r<BR>\n" );
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document.write( "* **Why the other options are incorrect:**<BR>\n" );
document.write( "    * **a) Biased sampling distribution:** This is incorrect. A valid confidence interval requires an *unbiased* sampling distribution for the sample mean.<BR>\n" );
document.write( "    * **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.<BR>\n" );
document.write( "    * **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. </SPAN>\n" );
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