Based on a smartphone survey, assume that 42% of adults with smartphones use them in theaters in a separate survey of 279 adults with smartphones, it is found that 116 use them in theaters a. if the 42% rate is correct, find the probability of getting 116 or fewer smartphone owners who use them in theaters b. Is the result of 116 significantly low? a. If the 42% rats is correct, the probability of getting 116 or fewer smartphone owners who use them in theaters is
To find the probability of getting 116 or fewer smartphone owners who use them in theaters, we will use the cumulative probability.
First, we need to calculate the probability of getting exactly 116 smartphone owners who use them in theaters.
P(X = 116) = (279 choose 116) * (0.42)^116 * (1-0.42)^(279-116)
Using a combination formula, (279 choose 116) = 279! / (116! * (279-116)!) = 2.622693e+50
P(X = 116) = 2.622693e+50 * (0.42)^116 * (0.58)^163 ≈ 1.811311e-30
Next, we need to calculate the cumulative probability of getting 116 or fewer smartphone owners who use them in theaters.
P(X ≤ 116) = P(X = 0) + P(X = 1) + ... + P(X = 116)
Using a cumulative probability calculator or a statistical software, we can find this probability to be approximately 5.77003e-24.
Therefore, the probability of getting 116 or fewer smartphone owners who use them in theaters, if the 42% rate is correct, is approximately 5.77003e-24.
b. To determine if the result of 116 is significantly low, we can compare it to a predetermined significance level (e.g., α = 0.05).
If the p-value (probability) is less than the significance level, we reject the null hypothesis (that the rate is 42%) and conclude that the result is significantly low.
Since the probability of getting 116 or fewer smartphone owners who use them in theaters is extremely low (5.77003e-24), it is significantly lower than the significance level of 0.05. Therefore, we can conclude that the result of 116 is significantly low.
To find the probability of getting 116 or fewer smartphone owners who use them in theaters, we can use the binomial distribution formula.
Let's denote the probability of an adult using their smartphone in theaters as p. In this case, p = 42% or 0.42.
The formula for the probability distribution of a binomial random variable is:
P(X ≤ k) = ∑ [nCk * p^k * (1-p)^(n-k)]
Where:
- P(X ≤ k) represents the probability of getting k or fewer successes
- n is the sample size (279 in this case)
- k is the number of successes we want to calculate the probability for
- nCk is the binomial coefficient, which represents the number of ways to choose k objects from a set of n objects.
Using this formula, we can calculate the probability of getting 116 or fewer smartphone owners who use them in theaters:
P(X ≤ 116) = ∑ [279Ck * 0.42^k * (1-0.42)^(279-k)], where k ranges from 0 to 116.
The probability can be calculated by summing up these values.