Some car tires can develop what is known as "heel and toe" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was 25.84 (in 10^3 miles ) and the variance was 3.76 (in 10^6 miles^2). For the cars with brand B, the corresponding statistics were 24.64 (in 10^3 miles) and 8.80 (in 10^6 miles^2) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands. Part a) Calculate the pooled variance s^2 to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. ________________×10^6 miles^2 Part b) Determine a 95% confidence interval for μA−μB, the difference in the mean 10^3 mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. (_________ ,_________) Part c) Based on the 95% confidence interval constructed in the previous part, which of the following conclusions can be drawn when we test H0:μA=μB vs. Ha:μA≠μB with α=0.05. A. Do not reject H0 since 1.20 is within the interval found in part (b). B. Do not reject H0 since 0 is within the interval found in part (b). C. Reject H0 since 0 is not within the interval found in part (b). D. Reject H0 since 0 is in the interval found in part (b). E. Do not reject H0 since 0 is not in the interval found in part (b).
Addition Rule of Probability
It simply refers to the likelihood of an event taking place whenever the occurrence of an event is uncertain. The probability of a single event can be calculated by dividing the number of successful trials of that event by the total number of trials.
Expected Value
When a large number of trials are performed for any random variable ‘X’, the predicted result is most likely the mean of all the outcomes for the random variable and it is known as expected value also known as expectation. The expected value, also known as the expectation, is denoted by: E(X).
Probability Distributions
Understanding probability is necessary to know the probability distributions. In statistics, probability is how the uncertainty of an event is measured. This event can be anything. The most common examples include tossing a coin, rolling a die, or choosing a card. Each of these events has multiple possibilities. Every such possibility is measured with the help of probability. To be more precise, the probability is used for calculating the occurrence of events that may or may not happen. Probability does not give sure results. Unless the probability of any event is 1, the different outcomes may or may not happen in real life, regardless of how less or how more their probability is.
Basic Probability
The simple definition of probability it is a chance of the occurrence of an event. It is defined in numerical form and the probability value is between 0 to 1. The probability value 0 indicates that there is no chance of that event occurring and the probability value 1 indicates that the event will occur. Sum of the probability value must be 1. The probability value is never a negative number. If it happens, then recheck the calculation.
Some car tires can develop what is known as "heel and toe" wear if not rotated after a certain mileage. To assess this issue, a consumer group investigated the tire wear on two brands of tire, A and B, say. Fifteen cars were fitted with new brand A tires and thirteen with brand B tires, the cars assigned to brand at random. (Two cars initially assigned to brand B suffered serious tire faults other than heel and toe wear, and were excluded from the study.) The cars were driven in regular driving conditions, and the mileage at which heal and toe wear could be observed was recorded on each car. For the cars with brand A tires, the mean mileage observed was 25.84 (in 10^3 miles ) and the variance was 3.76 (in 10^6 miles^2). For the cars with brand B, the corresponding statistics were 24.64 (in 10^3 miles) and 8.80 (in 10^6 miles^2) respectively. The mileage before heal and toe wear is detectable is assumed to be Normally distributed for both brands.
Part a) Calculate the pooled variance s^2 to 3 decimal places. During intermediate steps to arrive at the answer, make sure you keep as many decimal places as possible so that you can achieve the precision required in this question. ________________×10^6 miles^2
Part b) Determine a 95% confidence interval for μA−μB, the difference in the mean 10^3 mileages before heal and toe wear for the two brands of tire. Leave your answer to 2 decimal places. (_________ ,_________)
Part c) Based on the 95% confidence interval constructed in the previous part, which of the following conclusions can be drawn when we test H0:μA=μB vs. Ha:μA≠μB with α=0.05.
A. Do not reject H0 since 1.20 is within the interval found in part (b).
B. Do not reject H0 since 0 is within the interval found in part (b).
C. Reject H0 since 0 is not within the interval found in part (b).
D. Reject H0 since 0 is in the interval found in part (b).
E. Do not reject H0 since 0 is not in the interval found in part (b).
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