Questions: The average American gets a haircut every 38 days. Is the average smaller for college students? The data below shows the results of a survey of 16 college students asking them how many days elapse between haircuts. Assume that the distribution of the population is normal. 43,44,30,38,35,26,27,39, 26,24,34,33,30,26,30,38 The final conclusion is a. The data suggest the population mean is significantly lower than 38 at a=0.01, so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38. b. The data suggest the population mean number of days between haircuts for college students is not significantly lower than 38 at a=0.01, so there is insufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38. c. The data suggest the population mean is not significantly lower than 38 at α= 0.01 , so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38.

The average American gets a haircut every 38 days. Is the average smaller for college students? The data below shows the results of a survey of 16 college students asking them how many days elapse between haircuts. Assume that the distribution of the population is normal. 43,44,30,38,35,26,27,39, 26,24,34,33,30,26,30,38 The final conclusion is a. The data suggest the population mean is significantly lower than 38 at a=0.01, so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38. b. The data suggest the population mean number of days between haircuts for college students is not significantly lower than 38 at a=0.01, so there is insufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38. c. The data suggest the population mean is not significantly lower than 38 at α= 0.01 , so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38.
Transcript text: The average American gets a haircut every 38 days. Is the average smaller for college students? The data below shows the results of a survey of 16 college students asking them how many days elapse between haircuts. Assume that the distribution of the population is normal. $43,44,30,38,35,26,27,39$, $26,24,34,33,30,26,30,38$ The final conclusion is a. The data suggest the populaton mean is significantly lower than 38 at $a=0.01$, so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38. b. The data suggest the population mean number of days between haircuts for college students is not significantly lower than 38 at $a=0.01$, so there is insufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38. c.The data suggest the population mean is not significantly lower than 38 at $\alpha=$ 0.01 , so there is sufficient evidence to conclude that the population mean number of days between haircuts for college
failed

Solution

failed
failed

Solution Steps

To determine if the average number of days between haircuts for college students is significantly lower than 38 days, we can perform a one-sample t-test. The null hypothesis (H0) is that the mean number of days between haircuts for college students is 38 days. The alternative hypothesis (H1) is that the mean number of days is less than 38 days. We will use a significance level (alpha) of 0.01.

Solution Approach
  1. Calculate the sample mean and sample standard deviation of the given data.
  2. Perform a one-sample t-test comparing the sample mean to the population mean of 38 days.
  3. Determine the p-value from the t-test.
  4. Compare the p-value to the significance level (0.01) to decide whether to reject the null hypothesis.
Step 1: Calculate Sample Mean and Standard Deviation

Given the data: \[ 43, 44, 30, 38, 35, 26, 27, 39, 26, 24, 34, 33, 30, 26, 30, 38 \]

The sample mean (\(\bar{x}\)) is calculated as: \[ \bar{x} = 32.6875 \]

The sample standard deviation (\(s\)) is: \[ s = 6.3321 \]

Step 2: Perform One-Sample t-Test

We perform a one-sample t-test to compare the sample mean to the population mean (\(\mu_0 = 38\)).

The t-statistic is calculated using: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \]

Where \(n\) is the sample size (\(n = 16\)).

Step 3: Determine the p-Value

The p-value for the one-tailed test is: \[ p = 0.0022 \]

Step 4: Compare p-Value to Significance Level

We compare the p-value to the significance level (\(\alpha = 0.01\)).

Since \( p < \alpha \) and the t-statistic is negative, we reject the null hypothesis.

Final Answer

The data suggest the population mean is significantly lower than 38 at \(\alpha = 0.01\), so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38.

\[ \boxed{\text{a. The data suggest the population mean is significantly lower than 38 at } \alpha = 0.01, \text{ so there is sufficient evidence to conclude that the population mean number of days between haircuts for college students is lower than 38.}} \]

Was this solution helpful?
failed
Unhelpful
failed
Helpful

Questions asked by other users

failedAnswered
Which data scale is represented in this graph? a.) Ordinal b.) Nominal
failedAnswered
A cell with extra sets of chromosomes is said to be
failedAnswered
Determine the number of moles in each of the quantities below. a) 25 g NaCl b) 125 g H2SO4 c) 3,100 g KMnO4 d) 74 g KCl e) 35 g CuSO4
failedAnswered
Select all true statements about √56. It is a rational number. It is a real number. It is an integer. It is an irrational number. It is not a real number.
failedAnswered
The paradox of thrift demonstrates a delicate balance between individual financial prudence and its broader impact on the economy. When individuals or a household decides to save more (whether the reason be for financial security or a specific expense), this can unintentionally harm the overall economy. When people reduce their spending on goods and services, businesses experience decreased demand, potentially leading to layoffs and lower incomes. An example of the paradox of thrift is when an individual decides to save money by cooking at home instead of dining out. This seemingly sensible choice, while beneficial at a personal level, has broader repercussions. As fewer people frequent restaurants, the demand for their dishes decreases, leading to reduced production. Consequently, both restaurant owners and employees experience lower incomes. The reverse paradox of thrift is when people increase spending rather than saving. In theory, this boosts aggregate demand, stimulating production and the economy. However, caution is essential as overspending can lead to debt accumulation and potential financial instability. Moreover, excessive spending without adequate saving may also contribute to larger economic issues, such as inflation. Reflecting on the paradox of thrift, I realize that my focus on personal finances often overshadowed its wider impact. Previously, I considered only my household budget without fully appreciating how individual saving choices ripple through businesses and the economy. It's a valuable lesson in balancing saving and