A Simple Random Sample Of $n=17$ Is Drawn From A Population That Is Normally Distributed. The Sample Mean Is Found To Be $\bar{x}=50$, And The Sample Standard Deviation Is Found To Be \$s=19$[/tex\]. Construct A 95%

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A Simple Random Sample: Constructing a 95% Confidence Interval

In statistics, a simple random sample is a subset of individuals selected from a larger population, where every individual has an equal chance of being selected. This type of sampling is used to make inferences about the population based on the sample data. In this article, we will discuss how to construct a 95% confidence interval for a population mean using a simple random sample.

We are given a simple random sample of $n=17$ from a population that is normally distributed. The sample mean is found to be $\bar{x}=50$, and the sample standard deviation is found to be $s=19$. Our goal is to construct a 95% confidence interval for the population mean.

To construct a confidence interval for the population mean, we use the following formula:

xˉ±(Zα/2×sn)\bar{x} \pm (Z_{\alpha/2} \times \frac{s}{\sqrt{n}})

where:

  • \bar{x}$ is the sample mean

  • s$ is the sample standard deviation

  • n$ is the sample size

  • Z_{\alpha/2}$ is the critical value from the standard normal distribution for a given confidence level

For a 95% confidence interval, we need to find the critical value $Z_{\alpha/2}$ from the standard normal distribution. The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. The critical value $Z_{\alpha/2}$ is the value that separates the area under the standard normal curve into two equal parts, with the area to the left of the critical value representing $\alpha/2$ of the total area.

Using a standard normal distribution table or calculator, we find that the critical value $Z_{\alpha/2}$ for a 95% confidence interval is approximately 1.96.

The margin of error is the amount by which the sample mean is expected to differ from the population mean. It is calculated as:

sn\frac{s}{\sqrt{n}}

In this case, the margin of error is:

1917≈4.23\frac{19}{\sqrt{17}} \approx 4.23

Now that we have the critical value and the margin of error, we can construct the 95% confidence interval for the population mean:

50±(1.96×4.23)50 \pm (1.96 \times 4.23)

50±8.2950 \pm 8.29

Therefore, the 95% confidence interval for the population mean is:

(41.71,58.29)(41.71, 58.29)

The 95% confidence interval gives us a range of values within which we expect the population mean to lie. In this case, we are 95% confident that the population mean lies between 41.71 and 58.29.

In this article, we constructed a 95% confidence interval for the population mean using a simple random sample. We used the formula for the confidence interval, found the critical value from the standard normal distribution, calculated the margin of error, and constructed the confidence interval. The 95% confidence interval gives us a range of values within which we expect the population mean to lie.

There are several limitations to this article. First, we assumed that the population is normally distributed, which may not be the case in reality. Second, we used a simple random sample, which may not be representative of the population. Finally, we used a 95% confidence interval, which may not be the desired level of confidence.

Future research could involve:

  • Using a different type of sampling method, such as stratified sampling or cluster sampling
  • Using a different level of confidence, such as 90% or 99%
  • Using a different type of distribution, such as a t-distribution or a non-normal distribution
  • Using a different type of data, such as categorical data or time-series data
  • [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. E., & Farber, B. A. (2018). Elementary statistics: Picturing the world. Cengage Learning.
  • [3] Agresti, A., & Franklin, C. A. (2018). Statistics: The art and science of learning from data. Pearson Education.
    A Simple Random Sample: Q&A

In our previous article, we discussed how to construct a 95% confidence interval for a population mean using a simple random sample. In this article, we will answer some frequently asked questions about simple random sampling and confidence intervals.

A: A simple random sample is a subset of individuals selected from a larger population, where every individual has an equal chance of being selected. This type of sampling is used to make inferences about the population based on the sample data.

A: The assumptions of simple random sampling are:

  • The population is normally distributed
  • The sample is randomly selected from the population
  • The sample size is sufficiently large

A: The formula for the confidence interval is:

xˉ±(Zα/2×sn)\bar{x} \pm (Z_{\alpha/2} \times \frac{s}{\sqrt{n}})

where:

  • \bar{x}$ is the sample mean

  • s$ is the sample standard deviation

  • n$ is the sample size

  • Z_{\alpha/2}$ is the critical value from the standard normal distribution for a given confidence level

A: The confidence level is the probability that the confidence interval contains the population mean. Common confidence levels include 90%, 95%, and 99%. The choice of confidence level depends on the research question and the desired level of precision.

A: The margin of error is the amount by which the sample mean is expected to differ from the population mean. It is calculated as:

sn\frac{s}{\sqrt{n}}

A: The confidence interval gives us a range of values within which we expect the population mean to lie. For example, if the 95% confidence interval is (41.71, 58.29), we are 95% confident that the population mean lies between 41.71 and 58.29.

A: Some common mistakes to avoid when constructing a confidence interval include:

  • Not checking the assumptions of simple random sampling
  • Not using the correct formula for the confidence interval
  • Not choosing the correct confidence level
  • Not interpreting the confidence interval correctly

A: No, a confidence interval is used to make inferences about a population mean, not a population proportion. If you want to make inferences about a population proportion, you should use a confidence interval for proportions.

A: No, a confidence interval is used to make inferences about a population mean, not a population median. If you want to make inferences about a population median, you should use a confidence interval for medians.

In this article, we answered some frequently asked questions about simple random sampling and confidence intervals. We hope that this article has been helpful in clarifying some of the common misconceptions and misunderstandings about confidence intervals. If you have any further questions, please don't hesitate to ask.

  • [1] Moore, D. S., & McCabe, G. P. (2017). Introduction to the practice of statistics. W.H. Freeman and Company.
  • [2] Larson, R. E., & Farber, B. A. (2018). Elementary statistics: Picturing the world. Cengage Learning.
  • [3] Agresti, A., & Franklin, C. A. (2018). Statistics: The art and science of learning from data. Pearson Education.