How many patients were included in the random sample for the teeth straightening cost study?

  

The average cost for teeth straightening with metal braces is approximately $5400. A nationwide franchise thinks that its cost is below that figure. A random sample of 28 patients across the country had an average cost of $5250 with standard deviation of $629. At= 0.025, can it be concluded that the mean is less than $5400?

Introduction:
Teeth straightening is a common dental procedure that is often performed through the use of metal braces. One of the biggest concerns for patients seeking this treatment is the cost. The nationwide franchise in this study believes that their cost for teeth straightening with metal braces is below the national average of approximately $5400. In this study, we analyze a random sample of 28 patients to determine whether this claim can be supported.

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Description:
The purpose of this study is to determine whether the average cost for teeth straightening with metal braces at this nationwide franchise is indeed lower than the national average of $5400. A sample of 28 patients was selected at random from across the country, and their average cost for the procedure was found to be $5250 with a standard deviation of $629. To determine whether this sample is statistically significant, we perform a hypothesis test using an alpha level of 0.025.

Our null hypothesis is that the true mean cost for teeth straightening with metal braces at this franchise is equal to or greater than $5400. Our alternative hypothesis is that the true mean cost is less than $5400. Using a t-test for a single mean, we calculate the test statistic (t = -1.71) and find that the p-value is 0.0500. Since our alpha level is 0.025, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the mean cost for teeth straightening with metal braces at this franchise is less than $5400.

Objectives:
– To understand the average cost of teeth straightening with metal braces
– To determine if a nationwide franchise’s cost for teeth straightening is below the national average
– To learn how to conduct a hypothesis test using a random sample data

Learning Outcomes:
Upon completion of this content, learners will be able to:
– Explain the national average cost of teeth straightening with metal braces
– Discuss the cost of teeth straightening by a nationwide franchise
– Understand how to calculate the mean and standard deviation of a random sample
– Set up a hypothesis test and identify the null and alternative hypothesis
– Calculate the test statistic and determine the p-value
– Interpret the results of a hypothesis test
– Determine if the mean cost of teeth straightening by the nationwide franchise is below the national average with a significance level of 0.025.

Heading 1: Introduction
– Define average cost for teeth straightening with metal braces
– Explain the significance of determining if a nationwide franchise’s cost is lower than the national average

Heading 2: Sampling Method
– Discuss the random sample of 28 patients
– Explain how this sample size affects the accuracy of the results

Heading 3: Hypothesis Testing
– Define the null and alternative hypothesis
– Explain the significance level of 0.025
– Calculate the test statistic and the p-value
– Interpret the results of the hypothesis test

Heading 4: Conclusion
– Determine if we can conclude that the mean cost of teeth straightening by the nationwide franchise is lower than the national average
– Recap the main findings and the implications thereof.

Solution 1:

Using hypothesis testing, we can determine whether the nationwide franchise’s cost for teeth straightening with metal braces is below the average cost of $5400. Our null hypothesis is that the mean cost is $5400, while the alternative hypothesis is that the mean cost is less than $5400.

Using a t-test with a significance level of 0.025 and degrees of freedom (df) of 27 (n-1), we can determine the t-statistic and p-value for the sample mean of $5250. The t-statistic is calculated to be -1.228, while the p-value is 0.114.

Since the p-value is greater than the significance level, we fail to reject the null hypothesis. This means that there is not enough evidence to suggest that the mean cost for teeth straightening with metal braces at the nationwide franchise is less than $5400.

Solution 2:

Another possible solution is to calculate a confidence interval for the mean cost using the given sample data. The 95% confidence interval can be calculated as follows:

Lower bound = sample mean – (critical value)*(standard error)
Upper bound = sample mean + (critical value)*(standard error)

The critical value can be found using a t-distribution with df = 27 and a confidence level of 95%, which gives a value of 2.056. The standard error can be calculated as the sample standard deviation divided by the square root of the sample size, which gives a value of 118.88.

Plugging in the values, we get a 95% confidence interval of $4967.57 to $5532.43. Since this interval includes the value of $5400, we can conclude that there is not enough evidence to suggest that the mean cost for teeth straightening with metal braces at the nationwide franchise is below $5400.

Suggested Resources/Books:
1. Basic Statistics for Business and Economics by Douglas A. Lind, William G. Marchal, Samuel A. Wathen
2. Statistics for People Who (Think They) Hate Statistics by Neil J. Salkind
3. Statistics in Plain English by Timothy C. Urdan
4. Statistical Techniques in Business and Economics by Douglas A. Lind, William G. Marchal, Samuel A. Wathen

Similar Asked Questions:
1. What is the average cost of teeth straightening?
2. What is the standard deviation of the cost of teeth straightening?
3. Are the costs of teeth straightening for metal braces and Invisalign similar?
4. Is the cost of teeth straightening covered by insurance?
5. What factors affect the cost of teeth straightening?The average cost for teeth straightening with metal braces is approximately $5400. A nationwide franchise thinks that its cost is below that figure. A random sample of 28 patients across the country had an average cost of $5250 with standard deviation of $629. At= 0.025, can it be concluded that the mean is less than $5400?

Introduction:
Teeth straightening is a common dental procedure that is often performed through the use of metal braces. One of the biggest concerns for patients seeking this treatment is the cost. The nationwide franchise in this study believes that their cost for teeth straightening with metal braces is below the national average of approximately $5400. In this study, we analyze a random sample of 28 patients to determine whether this claim can be supported.

Description:
The purpose of this study is to determine whether the average cost for teeth straightening with metal braces at this nationwide franchise is indeed lower than the national average of $5400. A sample of 28 patients was selected at random from across the country, and their average cost for the procedure was found to be $5250 with a standard deviation of $629. To determine whether this sample is statistically significant, we perform a hypothesis test using an alpha level of 0.025.

Our null hypothesis is that the true mean cost for teeth straightening with metal braces at this franchise is equal to or greater than $5400. Our alternative hypothesis is that the true mean cost is less than $5400. Using a t-test for a single mean, we calculate the test statistic (t = -1.71) and find that the p-value is 0.0500. Since our alpha level is 0.025, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that the mean cost for teeth straightening with metal braces at this franchise is less than $5400.

Objectives:
– To understand the average cost of teeth straightening with metal braces
– To determine if a nationwide franchise’s cost for teeth straightening is below the national average
– To learn how to conduct a hypothesis test using a random sample data

Learning Outcomes:
Upon completion of this content, learners will be able to:
– Explain the national average cost of teeth straightening with metal braces
– Discuss the cost of teeth straightening by a nationwide franchise
– Understand how to calculate the mean and standard deviation of a random sample
– Set up a hypothesis test and identify the null and alternative hypothesis
– Calculate the test statistic and determine the p-value
– Interpret the results of a hypothesis test
– Determine if the mean cost of teeth straightening by the nationwide franchise is below the national average with a significance level of 0.025.

Heading 1: Introduction
– Define average cost for teeth straightening with metal braces
– Explain the significance of determining if a nationwide franchise’s cost is lower than the national average

Heading 2: Sampling Method
– Discuss the random sample of 28 patients
– Explain how this sample size affects the accuracy of the results

Heading 3: Hypothesis Testing
– Define the null and alternative hypothesis
– Explain the significance level of 0.025
– Calculate the test statistic and the p-value
– Interpret the results of the hypothesis test

Heading 4: Conclusion
– Determine if we can conclude that the mean cost of teeth straightening by the nationwide franchise is lower than the national average
– Recap the main findings and the implications thereof.

Solution 1:

Using hypothesis testing, we can determine whether the nationwide franchise’s cost for teeth straightening with metal braces is below the average cost of $5400. Our null hypothesis is that the mean cost is $5400, while the alternative hypothesis is that the mean cost is less than $5400.

Using a t-test with a significance level of 0.025 and degrees of freedom (df) of 27 (n-1), we can determine the t-statistic and p-value for the sample mean of $5250. The t-statistic is calculated to be -1.228, while the p-value is 0.114.

Since the p-value is greater than the significance level, we fail to reject the null hypothesis. This means that there is not enough evidence to suggest that the mean cost for teeth straightening with metal braces at the nationwide franchise is less than $5400.

Solution 2:

Another possible solution is to calculate a confidence interval for the mean cost using the given sample data. The 95% confidence interval can be calculated as follows:

Lower bound = sample mean – (critical value)*(standard error)
Upper bound = sample mean + (critical value)*(standard error)

The critical value can be found using a t-distribution with df = 27 and a confidence level of 95%, which gives a value of 2.056. The standard error can be calculated as the sample standard deviation divided by the square root of the sample size, which gives a value of 118.88.

Plugging in the values, we get a 95% confidence interval of $4967.57 to $5532.43. Since this interval includes the value of $5400, we can conclude that there is not enough evidence to suggest that the mean cost for teeth straightening with metal braces at the nationwide franchise is below $5400.

Suggested Resources/Books:
1. Basic Statistics for Business and Economics by Douglas A. Lind, William G. Marchal, Samuel A. Wathen
2. Statistics for People Who (Think They) Hate Statistics by Neil J. Salkind
3. Statistics in Plain English by Timothy C. Urdan
4. Statistical Techniques in Business and Economics by Douglas A. Lind, William G. Marchal, Samuel A. Wathen

Similar Asked Questions:
1. What is the average cost of teeth straightening?
2. What is the standard deviation of the cost of teeth straightening?
3. Are the costs of teeth straightening for metal braces and Invisalign similar?
4. Is the cost of teeth straightening covered by insurance?
5. What factors affect the cost of teeth straightening?

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