Recall that small effects may be statistically significant if the samples are large. A study of small-business failures looked at 153 food-and-drink businesses. Of these, 107 were headed by men and 46 were headed by women. During a three-year period, 17 of the men's businesses and 7 of the women's businesses failed. (a) Find the proportions of failures for businesses headed by men (sample 1) and businesses headed by women (sample 2). These sample proportions are quite close to each other. p̂men = p̂women = Give the P-value for the z test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative.) The test is very far from being significant. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = (b) Now suppose that the same sample proportions came from a sample of 30 times as large. That is, 210 out of 1380 business headed by women and 510 out of 3210 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in (a). Repeat the z test for the new data, and show that it is now more significant. (Round your test statistic to two decimal places and your P-value to four decimal places.) z = P-value = (c) Give the 95% confidence intervals for the difference between the proportions of men's and women's businesses that fail from Part (a) and Part (b). For part (a): 95% CI = , For part (b): 95% CI =
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.
| p̂men | = |
| p̂women | = |
Give the P-value for the z test of the hypothesis that the same proportion of women's and men's businesses fail. (Use the two-sided alternative.) The test is very far from being significant. (Round your test statistic to two decimal places and your P-value to four decimal places.)
| z | = |
| P-value | = |
(b) Now suppose that the same sample proportions came from a sample of 30 times as large. That is, 210 out of 1380 business headed by women and 510 out of 3210 businesses headed by men fail. Verify that the proportions of failures are exactly the same as in (a). Repeat the z test for the new data, and show that it is now more significant. (Round your test statistic to two decimal places and your P-value to four decimal places.)
| z | = |
| P-value | = |
(c) Give the 95% confidence intervals for the difference between the proportions of men's and women's businesses that fail from Part (a) and Part (b).
| For part (a): | ||
| 95% CI =
|
| For part (b): | ||
| 95% CI =
|
(d) What is the effect of larger samples on the confidence interval?
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