Before i state my question I would like to note that I have to do a one tailed test but i don't know how to with the information given. I need a standard deviation, i need a mean, but do i use the mean for yrs drivin or number accidents. Please explain
2.) Car insurance companies assume that the longer a person has been driving, the less likely they will be in an accident, and therefore charge new drivers higher insurance premiums than experienced drivers. To determine whether driving experience is related to the amount of car accidents, you survey a random sample of 12 Torontonians and ask them about the number of years they have been driving, and the number of car accidents they have been involved in during the past year. The data are presented below:
Driver #ofyrsdriving(X) #accidents (Y)
A 4.5 3
B 2.5 5
C 1.5 3
D 3 3
E 1.5 6
F 5 2
G 5 0
H 2 4
I 3 1
J 4 2
K 1 5
L 3 2
a. Determine whether the assumptions of car insurance companies are valid. Assuming á=0.05, include the hypotheses, critical test statistic, conclusion, and all formulas and calculations.
b. Is it appropriate to conclude that lack of driving experience causes accidents? Why or why not?
To determine whether the assumptions of car insurance companies are valid in this case, you will need to conduct a statistical hypothesis test. In order to do that, you will need both the mean and standard deviation for the number of years driving and the number of accidents.
In this scenario, you are interested in determining whether driving experience (number of years driving) is related to the number of car accidents. Therefore, you will use the mean and standard deviation for the number of years driving.
a. Hypotheses:
Null hypothesis (H0): There is no relationship between the number of years driving and the number of car accidents.
Alternative hypothesis (Ha): There is a relationship between the number of years driving and the number of car accidents.
b. Test statistic:
To perform this hypothesis test, you can use a correlation coefficient (r) to measure the strength and direction of the linear relationship between the two variables. The correlation coefficient ranges from -1 to +1, where -1 indicates a perfect negative relationship, +1 indicates a perfect positive relationship, and 0 indicates no linear relationship.
The critical test statistic depends on the sample size and the significance level (α) chosen for the test. In this case, α=0.05.
c. Calculations:
Step 1: Calculate the mean (X̄) and standard deviation (s) for the number of years driving (X):
X̄ = (4.5+2.5+1.5+3+1.5+5+5+2+3+4+1+3) / 12
= 38 / 12
= 3.17 years
s = √ [(Σ(X-X̄)^2) / (n-1)]
= √ [( (4.5-3.17)^2 + (2.5-3.17)^2 + ... + (3-3.17)^2 + ... + (3-3.17)^2 ) / (12-1)]
= √ [ (0.9996 + 0.2864 + 1.0904 + ... + 0.0289 + ... + 0.0289) / 11]
= √ [10.0461 / 11]
= √ [0.9133]
≈ 0.9556
Step 2: Calculate the correlation coefficient (r) between the number of years driving (X) and the number of accidents (Y):
r = Σ[(X-X̄)(Y-Ȳ)] / [(n-1)sXsY]
= [(4.5-3.17)(3-2.33) + (2.5-3.17)(5-2.33) + ... + (3-3.17)(2-2.33)] / [(11)(0.9556)(1.1819)]
= [-0.8295 + 0.8623 + ... + -0.6370] / 12.742
= -0.0934 / 12.742
≈ -0.0073
Step 3: Calculate the critical test statistic:
For a correlation coefficient test with 10 degrees of freedom and a significance level of α=0.05, the critical t-value is approximately ±2.228.
Step 4: Decision:
Since the absolute value of the calculated correlation coefficient (|-0.0073|) is less than the critical t-value (|2.228|), we fail to reject the null hypothesis. There is not enough evidence to support a relationship between the number of years driving and the number of car accidents.
b. Conclusion:
Based on the hypothesis test, we cannot conclude that lack of driving experience causes accidents since there is no significant relationship found between the number of years driving and the number of car accidents in this sample.
To determine whether the assumptions of car insurance companies are valid in this scenario, you can perform a hypothesis test. The mean and standard deviation play important roles in this process.
a. Hypotheses:
To test the assumption that longer driving experience is related to a lower number of car accidents, we need to compare the means of the two variables (years driving and number of accidents). Let's consider the following hypotheses:
Null Hypothesis (H0): The mean number of car accidents is the same for all levels of driving experience (no relationship between driving experience and accident rate).
Alternative Hypothesis (Ha): The mean number of car accidents decreases as driving experience increases (driving experience is related to a lower accident rate).
Test statistic:
In order to perform the test, we can calculate a test statistic called the t-statistic. The t-statistic is given by the formula:
t = (mean difference - hypothesized mean difference) / (standard deviation / sqrt(sample size))
To calculate the mean difference, subtract the mean of the number of accidents from the mean of the years driving. The hypothesized mean difference is 0, assuming no difference between the two variables. The standard deviation will be required to perform this calculation.
Mean and standard deviation:
In this case, since we are comparing the means of two variables, we need to calculate the mean and standard deviation for both variables - years driving and number of accidents.
Mean of years driving (X):
To obtain the mean of years driving, sum up all the values of years driving and divide by the sample size (which is 12 in this case):
Mean of X = (4.5 + 2.5 + 1.5 + 3 + 1.5 + 5 + 5 + 2 + 3 + 4 + 1 + 3) / 12
Standard deviation of years driving (X):
The standard deviation is a measure of dispersion or spread around the mean. Calculate the sample standard deviation for years driving using the following formula:
Standard deviation of X = sqrt(sum((X - mean of X)^2) / (sample size - 1))
Mean of number of accidents (Y):
Similarly, calculate the mean of number of accidents using the given data:
Mean of Y = (3 + 5 + 3 + 3 + 6 + 2 + 0 + 4 + 1 + 2 + 5 + 2) / 12
Standard deviation of number of accidents (Y):
Calculate the sample standard deviation for number of accidents using the formula:
Standard deviation of Y = sqrt(sum((Y - mean of Y)^2) / (sample size - 1))
Once you have calculated the mean and standard deviation for both variables, you can proceed with the hypothesis test.
Hypothesis test:
Plug all the values into the test statistic formula and calculate the t-value. Compare the calculated t-value with the critical t-value from the t-distribution table (with (n-2) degrees of freedom, where n is the sample size). If the calculated t-value is greater than the critical t-value, we reject the null hypothesis; otherwise, we fail to reject the null hypothesis.
Conclude based on the decision made in the hypothesis test.
b. The conclusion from the hypothesis test will provide evidence either in support or against the assumption made by car insurance companies. However, it is important to note that correlation does not imply causation. Even if the hypothesis test supports the assumption that driving experience is related to a lower accident rate, it does not conclusively prove that lack of driving experience causes accidents. There could be other factors at play that contribute to the relationship between driving experience and accident rate.