Questions: In this assignment you will practice building and interpreting confidence intervals. Confidence intervals are used to give a range of plausible values for the population mean or proportion under consideration. If you have ever read the results of a poll, you might recall that a percentage is given followed by the statement, "...plus or minus..." some value. For example, the presidential approval rating is 49% plus or minus 1.73%. The 49% is the sample proportion with the 1.73% noting the margin of error. Understanding how confidence intervals are built and interpreted will help you become a better consumer of information. Confidence intervals are used in a variety of applications besides analyzing polling data. For instance, manufacturing clothing (sizes are based on an average plus or minus an acceptable error), volume of food items (the 12 oz cereal box holds on average 12 oz plus or minus a margin of error), lifetimes of items (a lightbulb will last on average 400 hours plus or minus an acceptable error), etc. Task Based on USDA suggestion, a U.S. adult should be spending anywhere from 165 to 345 per month on food, depending on age and gender. A random sample of 120 adults in Washington State is polled regarding the amount of money they spend each month on food. The sample has an average of 273.61 with a standard deviation of 61.59. Based on this information, complete the following questions. a) (5pts)Show that the requirements to build a confidence interval are met. b) (5 pts) State the critical value for a 90% confidence interval c) (5pts)Compute the 90% confidence interval. State values to two decimal places. d) (5pts) Interpret the 90% confidence interval using complete sentences.

In this assignment you will practice building and interpreting confidence intervals. Confidence intervals are used to give a range of plausible values for the population mean or proportion under consideration. If you have ever read the results of a poll, you might recall that a percentage is given followed by the statement, "...plus or minus..." some value. For example, the presidential approval rating is 49% plus or minus 1.73%. The 49% is the sample proportion with the 1.73% noting the margin of error. Understanding how confidence intervals are built and interpreted will help you become a better consumer of information. Confidence intervals are used in a variety of applications besides analyzing polling data. For instance, manufacturing clothing (sizes are based on an average plus or minus an acceptable error), volume of food items (the 12 oz cereal box holds on average 12 oz plus or minus a margin of error), lifetimes of items (a lightbulb will last on average 400 hours plus or minus an acceptable error), etc. Task Based on USDA suggestion, a U.S. adult should be spending anywhere from 165 to 345 per month on food, depending on age and gender. A random sample of 120 adults in Washington State is polled regarding the amount of money they spend each month on food. The sample has an average of 273.61 with a standard deviation of 61.59. Based on this information, complete the following questions. a) (5pts)Show that the requirements to build a confidence interval are met. b) (5 pts) State the critical value for a 90% confidence interval c) (5pts)Compute the 90% confidence interval. State values to two decimal places. d) (5pts) Interpret the 90% confidence interval using complete sentences.
Transcript text: In this assignment you will practice building and interpreting confidence intervals. Confidence intervals are used to give a range of plausible values for the population mean or proportion under consideration. If you have ever read the results of a poll, you might recall that a percentage is given followed by the statement, "...plus or minus..." some value. For example, the presidential approval rating is $49 \%$ plus or minus $1.73 \%$. The $49 \%$ is the sample proportion with the $1.73 \%$ noting the margin of error Understanding how confidence intervals are built and interpreted will help you become a better consumer of information. Confidence intervals are used in a variety of applications besides analyzing polling data. For instance, manufacturing clothing (sizes are based on an average plus or minus an acceptable error), volume of food items (the 12 oz cereal box holds on average 12 oz plus or minus a margin of error), lifetimes of items (a lightbulb will last on average 400 hours plus or minus a n acceptable error), etc. Task Based on USDA suggestion, a U.S. adult should be spending anywhere from $\$ 165$ to $\$ 345$ per month on food, depending on age and gender. A random sample of 120 adults in Washington State is polled regarding the amount of money they spend each month on food. The sample has an average of \$273.61 with a standard deviation of $\$ 61.59$. Based on this information, complete the following questions. a) (5pts)Show that the requirements to build a confidence interval are met. b) $(5 \mathrm{pts})$ State the critical value for a $90 \%$ confidence interval c) (5pts)Compute the $90 \%$ confidence interval. State values to two decimal places. d) (5pts) Interpret the $90 \%$ confidence interval using complete sentences.
failed

Solution

failed
failed

Solution Steps

Step 1: Requirements for Confidence Interval

To build a confidence interval, the following requirements must be met:

  1. The sample should be random.
  2. The sample size should be large (\(n \geq 30\)) or the population should be normally distributed.

In this case, the sample size is \(n = 120\), which satisfies the requirement for a large sample size.

Step 2: Critical Value for 90% Confidence Interval

The critical value (\(z^*\)) for a 90% confidence interval can be found using the standard normal distribution. For a 90% confidence level, the critical value is approximately:

\[ z^* \approx 1.645 \]

Step 3: Compute the 90% Confidence Interval

The confidence interval for the mean is calculated using the formula:

\[ \bar{x} \pm z^* \frac{s}{\sqrt{n}} \]

Substituting the values:

\[ 273.61 \pm 1.645 \cdot \frac{61.59}{\sqrt{120}} \]

Calculating the margin of error:

\[ \text{Margin of Error} = 1.645 \cdot \frac{61.59}{\sqrt{120}} \approx 9.25 \]

Thus, the confidence interval is:

\[ (273.61 - 9.25, 273.61 + 9.25) = (264.36, 282.86) \]

Final Answer

The 90% confidence interval for the average amount of money spent per month on food by adults in Washington State is:

\[ \boxed{(264.36, 282.86)} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
(b) Σ from n=2 to ∞ of (-1)^n * 5/ln(n)
failedAnswered
"Affirmation codes" were signals sent by amateur radio operators in the 1910s.
failedAnswered
U A B 2 13 5 10 17 21 23 26
failedAnswered
Use the vertex and intercepts to sketch the graph of the quadratic function. Give the equation for the parabola's axis of symmetry. Use the parabola to identify the function's domain and range. f(x)=(x-3)^2+1 The axis of symmetry is x=3. Identify the function's domain. The domain is (-∞, ∞). Identify the function's range. The range is [1, ∞).
failedAnswered
Most calculators can find logarithms with base and base e. To find logarithms with different bases, we use the Select. To find log base y of 15, we write the following. (Round your answers to three decimal places.) log base 9 of 16 = log() / log() = Do we get the same answer if we perform the calculation in part (a) using ln in place of log? Yes, the result is the same. No, the result is not the same.