The diameter of a wire should be 8 cm. It follows a normal distribution and has a standard deviation of 0.01 cm.
A person tests each delivery by taking 5 samples, accepting the goods only if the mean is at most 8.01
a)
Find the probability of type 1 error?
b)
A hypothesis test is to be carried out with P(type I error) = P(type II error) and either one being at 0.1%.
What is the necessary sample size for this test? (answer in my book says it's n=17)
2)
A company will sell a new product if the proportion of customers that will buy it is at least 20%. It is decided that the null hypothesis is rejected if the sample proportion is at least 27%. Sample size = 100.
a) Find P(type I error)
b)
If P(do not reject the null hypothesis)must be at least 99% if the true proportion is 0.17 and
P(reject the null hypothesis)must be at least 99% if the true proportion is 0.27:
which sample size is required?
(My book's answer is n = or greater than 365)
I don't understand how to solve these kind of questions, I'd greatly appreciate if you could help explain !
Help wanted in this subject
Sure, I can help you understand how to solve these types of questions. Let's start with the first one:
1a) To find the probability of a type 1 error, we first need to understand what it means. In hypothesis testing, a type 1 error occurs when we reject the null hypothesis even though it is true. In this case, the null hypothesis is that the mean diameter of the wire is 8 cm. The alternative hypothesis is that the mean diameter is greater than 8 cm.
Given that the wire diameter follows a normal distribution with a standard deviation of 0.01 cm, we can use the z-distribution to calculate the probability of a type 1 error.
To do this, we need to find the Z-score corresponding to a mean of 8.01 cm, if the null hypothesis is true. The Z-score formula is (X - μ) / σ, where X is the sample mean, μ is the population mean, and σ is the standard deviation. In this case, X = 8.01, μ = 8, and σ = 0.01.
The Z-score can be calculated as (8.01 - 8) / 0.01 = 0.01 / 0.01 = 1.
Next, we need to find the probability corresponding to a Z-score of 1 using a Z-table or calculator. The corresponding probability for a Z-score of 1 is approximately 0.8413.
Therefore, the probability of a type 1 error is 0.8413.
1b) In this case, we are given that the probability of a type 1 error (P(type 1 error)) is equal to the probability of a type 2 error (P(type 2 error)), and both are 0.1%.
A type 2 error occurs when we fail to reject the null hypothesis even though it is false. In this case, it would mean that the mean diameter is actually greater than 8.01 cm, but we fail to reject the null hypothesis based on our sample mean.
To find the necessary sample size for this test, we need to use the concept of power. Power is the probability of correctly rejecting the null hypothesis when it is false, and it is equal to 1 - P(type 2 error).
Given that P(type 2 error) = 0.1%, we can calculate the power as 1 - 0.1% = 99.9%.
To find the necessary sample size, we can use a statistical power calculator or formulas specific to the test being performed. Without additional information about the test being used, it is difficult to determine the exact calculations required. However, a sample size of n = 17 may be obtained using power calculations based on a specific effect size and significance level.
Moving on to the second set of questions:
2a) To find the probability of a type 1 error, we need to understand the given information. The null hypothesis is that the proportion of customers who will buy the new product is less than 20%. The alternative hypothesis is that the proportion is at least 20%.
Given that the sample size is 100 and the null hypothesis is rejected if the sample proportion is at least 27%, we can calculate the probability of a type 1 error.
Assuming the null hypothesis is true, the sample proportion follows a binomial distribution with parameters n = 100 (sample size) and π = 0.2 (proportion under the null hypothesis).
To calculate the probability of a type 1 error, we need to find the probability of observing a sample proportion of at least 27% (0.27) when the true proportion is 20% (0.20) or less.
Using a binomial distribution calculator or software, we can find the probability of observing a sample proportion of at least 0.27 when the true proportion is 0.20 or less. This probability is the probability of a type 1 error.
2b) In this case, we are given specific requirements for the probabilities of not rejecting the null hypothesis and rejecting the null hypothesis, depending on the true proportion.
If the true proportion is 0.17, we want the probability of not rejecting the null hypothesis to be at least 99%. Similarly, if the true proportion is 0.27, we want the probability of rejecting the null hypothesis to be at least 99%.
To determine the sample size required to meet these requirements, we need to find the minimum sample size that satisfies both conditions.
Using the concept of power, we can calculate the probability of rejecting the null hypothesis when it is false (1 - P(type 2 error)). In this case, the null hypothesis assumes a proportion of less than 20%, while the alternate hypothesis assumes a proportion of 0.27.
To meet the requirement of a 99% probability of rejecting the null hypothesis if the true proportion is 0.27, we need to ensure the power is at least 99%. Similarly, to meet the requirement of a 99% probability of not rejecting the null hypothesis if the true proportion is 0.17, the power should be less than or equal to 1 - 99% = 1%.
To determine the required sample size, we would need to perform power calculations specific to the test being used. The result of n ≥ 365 may be obtained using calculations based on specific effect sizes and significance levels.
I hope this helps clarify how to solve these types of questions. If you have any further questions, please let me know!