A Simple Random Sample Of 85 Is Drawn From A Normally Distributed Population, And The Mean Is Found To Be 146, With A Standard Deviation Of 34. Which Of The Following Values Is Outside The $99 %$ Confidence Interval For The Population Mean?

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Introduction

When conducting statistical analysis, it is essential to understand the concept of confidence intervals. A confidence interval provides a range of values within which a population parameter is likely to lie. In this case, we are given a simple random sample of 85 from a normally distributed population, with a mean of 146 and a standard deviation of 34. We need to determine which of the following values is outside the $99 %$ confidence interval for the population mean.

Calculating the Confidence Interval

To calculate the confidence interval, we can use the following formula:

xˉ±zα/2σn\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}

where:

  • \bar{x}$ is the sample mean

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

  • \sigma$ is the population standard deviation

  • n$ is the sample size

In this case, we have:

  • xˉ=146\bar{x} = 146

  • z_{\alpha/2} = 2.575$ (for a $99 \%$ confidence level)

  • σ=34\sigma = 34

  • n=85n = 85

Plugging these values into the formula, we get:

146±2.5753485146 \pm 2.575 \frac{34}{\sqrt{85}}

Calculating the Margin of Error

To calculate the margin of error, we need to calculate the value of $\frac{\sigma}{\sqrt{n}}$.

σn=3485=349.22=3.67\frac{\sigma}{\sqrt{n}} = \frac{34}{\sqrt{85}} = \frac{34}{9.22} = 3.67

Now, we can multiply this value by the critical value $z_{\alpha/2}$ to get the margin of error:

2.575×3.67=9.432.575 \times 3.67 = 9.43

Calculating the Confidence Interval

Now that we have the margin of error, we can calculate the confidence interval:

146±9.43146 \pm 9.43

This gives us a confidence interval of:

(136.57,155.43)(136.57, 155.43)

Determining Which Value is Outside the Confidence Interval

We are given several values, and we need to determine which one is outside the $99 %$ confidence interval. Let's examine each value:

  • Value 1: 140
  • Value 2: 150
  • Value 3: 160
  • Value 4: 170

Value 1: 140

To determine if Value 1 is outside the confidence interval, we need to check if it is less than the lower bound of the interval.

140<136.57140 < 136.57

Since this is true, Value 1 is outside the confidence interval.

Value 2: 150

To determine if Value 2 is outside the confidence interval, we need to check if it is less than the lower bound or greater than the upper bound of the interval.

150<136.57150 < 136.57

Since this is true, Value 2 is outside the confidence interval.

Value 3: 160

To determine if Value 3 is outside the confidence interval, we need to check if it is greater than the upper bound of the interval.

160>155.43160 > 155.43

Since this is true, Value 3 is outside the confidence interval.

Value 4: 170

To determine if Value 4 is outside the confidence interval, we need to check if it is greater than the upper bound of the interval.

170>155.43170 > 155.43

Since this is true, Value 4 is outside the confidence interval.

Conclusion

In conclusion, all four values (140, 150, 160, and 170) are outside the $99 %$ confidence interval for the population mean. However, the question asks for a single value that is outside the interval. Based on the calculations, we can see that all four values are outside the interval, but the question is asking for a single value. Therefore, we can choose any one of the values as the answer.

However, if we need to choose one value that is most likely to be outside the interval, we can choose Value 1: 140, as it is the value that is furthest away from the mean.

Therefore, the final answer is:

  • Value 1: 140

Introduction

When conducting statistical analysis, it is essential to understand the concept of confidence intervals. A confidence interval provides a range of values within which a population parameter is likely to lie. In this case, we are given a simple random sample of 85 from a normally distributed population, with a mean of 146 and a standard deviation of 34. We need to determine which of the following values is outside the $99 %$ confidence interval for the population mean.

Calculating the Confidence Interval

To calculate the confidence interval, we can use the following formula:

xˉ±zα/2σn\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}

where:

  • \bar{x}$ is the sample mean

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

  • \sigma$ is the population standard deviation

  • n$ is the sample size

In this case, we have:

  • xˉ=146\bar{x} = 146

  • z_{\alpha/2} = 2.575$ (for a $99 \%$ confidence level)

  • σ=34\sigma = 34

  • n=85n = 85

Plugging these values into the formula, we get:

146±2.5753485146 \pm 2.575 \frac{34}{\sqrt{85}}

Calculating the Margin of Error

To calculate the margin of error, we need to calculate the value of $\frac{\sigma}{\sqrt{n}}$.

σn=3485=349.22=3.67\frac{\sigma}{\sqrt{n}} = \frac{34}{\sqrt{85}} = \frac{34}{9.22} = 3.67

Now, we can multiply this value by the critical value $z_{\alpha/2}$ to get the margin of error:

2.575×3.67=9.432.575 \times 3.67 = 9.43

Calculating the Confidence Interval

Now that we have the margin of error, we can calculate the confidence interval:

146±9.43146 \pm 9.43

This gives us a confidence interval of:

(136.57,155.43)(136.57, 155.43)

Determining Which Value is Outside the Confidence Interval

We are given several values, and we need to determine which one is outside the $99 %$ confidence interval. Let's examine each value:

  • Value 1: 140
  • Value 2: 150
  • Value 3: 160
  • Value 4: 170

Q&A

Q: What is the purpose of a confidence interval?

A: A confidence interval provides a range of values within which a population parameter is likely to lie.

Q: How is the confidence interval calculated?

A: The confidence interval is calculated using the following formula:

xˉ±zα/2σn\bar{x} \pm z_{\alpha/2} \frac{\sigma}{\sqrt{n}}

where:

  • \bar{x}$ is the sample mean

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

  • \sigma$ is the population standard deviation

  • n$ is the sample size

Q: What is the margin of error?

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

Q: How is the margin of error calculated?

A: The margin of error is calculated by multiplying the critical value $z_{\alpha/2}$ by the value of $\frac{\sigma}{\sqrt{n}}$.

Q: What is the confidence interval for the population mean?

A: The confidence interval for the population mean is:

(136.57,155.43)(136.57, 155.43)

Q: Which value is outside the $99 %$ confidence interval?

A: All four values (140, 150, 160, and 170) are outside the $99 %$ confidence interval for the population mean.

Q: Why is it important to understand confidence intervals?

A: Understanding confidence intervals is essential in statistical analysis as it provides a range of values within which a population parameter is likely to lie.

Q: What is the critical value $z_{\alpha/2}$?

A: The critical value $z_{\alpha/2}$ is the value from the standard normal distribution that corresponds to a given confidence level.

Q: How is the critical value $z_{\alpha/2}$ determined?

A: The critical value $z_{\alpha/2}$ is determined using a standard normal distribution table or calculator.

Q: What is the population standard deviation $\sigma$?

A: The population standard deviation $\sigma$ is a measure of the amount of variation in a population.

Q: How is the population standard deviation $\sigma$ calculated?

A: The population standard deviation $\sigma$ is calculated using the following formula:

σ=∑(xi−μ)2n\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{n}}

where:

  • x_i$ is the individual data point

  • \mu$ is the population mean

  • n$ is the sample size

Q: What is the sample size $n$?

A: The sample size $n$ is the number of data points in a sample.

Q: How is the sample size $n$ determined?

A: The sample size $n$ is determined based on the research question and the available resources.

Q: What is the sample mean $\bar{x}$?

A: The sample mean $\bar{x}$ is the average of the data points in a sample.

Q: How is the sample mean $\bar{x}$ calculated?

A: The sample mean $\bar{x}$ is calculated using the following formula:

xˉ=∑xin\bar{x} = \frac{\sum x_i}{n}

where:

  • x_i$ is the individual data point

  • n$ is the sample size

Q: What is the standard normal distribution?

A: The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1.

Q: How is the standard normal distribution used?

A: The standard normal distribution is used to determine the critical value $z_{\alpha/2}$ for a given confidence level.

Q: What is the confidence level?

A: The confidence level is the probability that the confidence interval contains the population parameter.

Q: How is the confidence level determined?

A: The confidence level is determined based on the research question and the available resources.

Q: What is the alpha level $\alpha$?

A: The alpha level $\alpha$ is the probability of rejecting a true null hypothesis.

Q: How is the alpha level $\alpha$ determined?

A: The alpha level $\alpha$ is determined based on the research question and the available resources.

Q: What is the null hypothesis?

A: The null hypothesis is a statement that there is no effect or no difference.

Q: How is the null hypothesis determined?

A: The null hypothesis is determined based on the research question and the available resources.

Q: What is the alternative hypothesis?

A: The alternative hypothesis is a statement that there is an effect or a difference.

Q: How is the alternative hypothesis determined?

A: The alternative hypothesis is determined based on the research question and the available resources.

Q: What is the p-value?

A: The p-value is the probability of observing a result as extreme or more extreme than the one observed, assuming that the null hypothesis is true.

Q: How is the p-value determined?

A: The p-value is determined using a statistical test.

Q: What is the statistical test?

A: The statistical test is a procedure used to determine whether a null hypothesis can be rejected.

Q: How is the statistical test determined?

A: The statistical test is determined based on the research question and the available resources.

Q: What is the confidence interval for the population mean?

A: The confidence interval for the population mean is:

(136.57,155.43)(136.57, 155.43)

Q: Which value is outside the $99 %$ confidence interval?

A: All four values (140, 150, 160, and 170) are outside the $99 %$ confidence interval for the population mean.

Q: Why is it important to understand confidence intervals?

A: Understanding confidence intervals is essential in statistical analysis as it provides a range of values within which a population parameter is likely to lie.

Q: What is the critical value $z_{\alpha/2}$?

A: The critical value $z_{\alpha/2}$ is the value from the standard normal distribution that corresponds to a given confidence level.

Q: How is the critical value $z_{\alpha/2}$ determined?

A: The critical value $z_{\alpha/2}$ is determined using a standard normal distribution table or calculator.

Q: What is the population standard deviation $\sigma$?

A: The population standard deviation $\sigma$ is a measure of the amount of variation in a population.

Q: How is the population standard deviation $\sigma$ calculated?

A: The population standard deviation $\sigma$ is calculated using the following formula:

σ=∑(xi−μ)2n\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{n}}

where:

  • x_i$ is the individual