Hi there, I am doing my statistics homework and am having trouble with a few problems. I sincerely appreciate any help I receive! :)
According to a local chamber of commerce, in 1993, 5.9% of local area residents owned more than five cars. A local car dealer claims that the percentage has increases. He randomly selects 180 local area residents and finds that 12 of them own more than five cars. Test this car dealer’s claim that the (alpha = .05) level of significance.
Use the binomial theorem.
Show that you can’t use z.
To test the car dealer's claim using the binomial theorem, we need to set up the null and alternative hypotheses.
Null hypothesis (H0): The percentage of local area residents owning more than five cars has not changed.
Alternative hypothesis (Ha): The percentage of local area residents owning more than five cars has increased.
Now, let's calculate the expected number of local area residents who own more than five cars under the assumption that the null hypothesis is true.
The expected number can be calculated using the formula: E = n * p
Where:
E = expected number of successes (people owning more than five cars)
n = sample size (180 residents)
p = probability of success (the percentage determined by the chamber of commerce, which is 0.059)
Substituting the values into the formula, we get:
E = 180 * 0.059
E ≈ 10.62
Since the expected number of successes is greater than 5 (a general rule of thumb), we can proceed with using the binomial theorem.
The binomial probability formula can be used to find the probability of observing a specific number of successes (people owning more than five cars) in a given sample size (180 residents) under the assumption that the null hypothesis is true.
P(X = 12) = (nCr) * (p^x) * ((1-p)^(n-x))
Where:
X = number of successes (people owning more than five cars) in the sample
nCr = number of combinations of n items taken r at a time (using a calculator function such as nCr or the formula n! / (r! * (n - r)!))
p = probability of success (0.059)
x = specific number of successes (12)
n = sample size (180 residents)
Substituting the values into the formula, we get:
P(X = 12) = (180C12) * (0.059^12) * ((1 - 0.059)^(180-12))
Now you can calculate the value of P(X = 12) using a calculator or statistical software.
If the probability obtained is less than the specified level of significance (alpha = 0.05), then we reject the null hypothesis H0 in favor of the alternative hypothesis Ha. Otherwise, we fail to reject the null hypothesis.
Now, regarding the use of z, we cannot directly use the z-test because the sampling distribution in this case is not approximately normally distributed, which is a requirement for using the z-test. The binomial distribution, in this case, is discrete with two possible outcomes, so using the binomial theorem is more appropriate to analyze this situation.