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Chapter 3: Problem 45

Information about a sample is given. Assuming that the sampling distribution is symmetric and bell-shaped, use the information to give a \(95 \%\) confidence interval, and indicate the parameter being estimated. $$ \hat{p}=0.32 \text { and the standard error is } 0.04 . $$

Short Answer

Expert verified
The 95% confidence interval for the parameter \(p\) is [0.2416, 0.3984].

Step by step solution

01

Identify the Parameter Being Estimated

The parameter being estimated is \(p\), the proportion. We know that the point estimate for \(p\) is \(\hat{p}\) = 0.32 and that the standard error (SE) is 0.04.
02

Find the Standard Score for the Confidence Interval

To construct a 95% confidence interval, we need to find the value of the standard score (z) that leaves 5% in the tails of a standard normal distribution (since 100% - 95% = 5%). This is a common value and most statistics texts list it in a table. If looked up in the z-table, the value is approximately \(Z_{0.025}\) = 1.96 (since we need to look at the tail, we consider 伪/2 = 0.025).
03

Calculate the Confidence Interval

Once we have the standard score, we can construct the confidence interval. The 95% confidence interval is defined as follows: \(\hat{p} \pm Z_{0.025} \times SE\). Substituting the given values: \(0.32 \pm 1.96 \times 0.04\). Simplifying this gives us an interval from \(0.32 - 0.0784\) to \(0.32 + 0.0784\), or from \(0.2416\) to \(0.3984\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Standard Error
The concept of the Standard Error (SE) is central to understanding how sample statistics estimate population parameters. The Standard Error represents the average distance that the sample proportion (\( \hat{p} \)) is from the actual population proportion (\( p \)). It is calculated using the formula: \[ SE = \frac{s}{\sqrt{n}} \] where \( s \) is the standard deviation of the sample, and \( n \) is the sample size. In our problem, the SE is provided as 0.04.
  • This means that the sample's proportion estimate \( \hat{p} \) of 0.32 is expected to vary by 0.04 from the true population proportion.
  • A smaller SE suggests that our sample estimate is closer to the actual population parameter, increasing the accuracy and precision of our estimation.
Understanding SE is crucial because it helps us communicate the level of uncertainty or variability of our sample estimate. This forms the basis of constructing and interpreting confidence intervals.
Sampling Distribution
The Sampling Distribution refers to the probability distribution of a sample statistic. In our exercise, this concerns the sample proportion \( \hat{p} \).
It's assumed the sampling distribution is symmetric and bell-shaped, resembling the normal distribution. This assumption aligns with the Central Limit Theorem, which states that with a sufficiently large sample size, the sampling distribution of the sample mean or proportion will be approximately normally distributed, regardless of the population's original distribution.

  • The shape and spread of the sampling distribution depends on the sample size and variability within the sample.
  • Here, because the sampling distribution is symmetric and bell-shaped, we can use it to construct a confidence interval for \( p \) by expanding around the sample estimate \( \hat{p} \).
In essence, understanding sampling distribution allows us to predict how sample statistics like \( \hat{p} \) vary across different samples drawn from the same population.
Standard Score
The Standard Score, often referred to as the z-score, measures how many standard deviations a data point is from the mean. In the context of confidence intervals, it helps us understand where a sample proportion lies in relation to the center of the sampling distribution. For a 95% confidence interval, we identify the z-score that corresponds with leaving 5% in the tails of the normal distribution. From statistical tables, this often-used score is approximately 1.96. Here's how it's applied:
  • The formula \( \hat{p} \pm Z_{\alpha/2} \times SE \) uses the z-score to establish the range of values in which we expect the true parameter to lie with 95% confidence.
  • In our problem, \( \hat{p} = 0.32 \), SE = 0.04, and the z-score (1.96) extends the range around 0.32, offering an interval of 0.2416 to 0.3984.
Thus, the standard score is a crucial component in determining how well our sample describes the population, providing insight into the reliability of the estimated confidence interval.

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