A recent survey found that 63% of all adults over 50 wear glasses for driving. In a random sample of 80 adults over 50, what is the mean and standard deviation of those that wear glasses?
To find the mean and standard deviation of those that wear glasses, we first need to calculate the number of adults in the sample who wear glasses.
Given that 63% of all adults over 50 wear glasses, we can infer that the proportion of adults in the sample who wear glasses is also 63%. To find the number of adults who wear glasses in the sample, we can multiply this proportion by the sample size.
Number of adults who wear glasses = Proportion of adults wearing glasses * Sample size
Number of adults who wear glasses = 0.63 * 80 = 50.4 (rounding to the nearest whole number)
Since we cannot have a fraction of a person, we can conclude that approximately 50 adults in the sample wear glasses.
Now, using this information, we can calculate the mean and standard deviation.
Mean (average) = (Number of adults who wear glasses) / (Sample size)
Mean = 50 / 80 = 0.625 (rounded to three decimal places)
The mean or average proportion of adults over 50 who wear glasses is approximately 0.625.
To calculate the standard deviation, we need to use the formula for the standard error:
Standard deviation = sqrt( (p * q) / n)
where p = Proportion of adults who wear glasses, q = Proportion of adults who do not wear glasses (1 - p), and n = Sample size.
In this case, p = 0.63, q = 1 - 0.63 = 0.37, and n = 80.
Standard deviation = sqrt( (0.63 * 0.37) / 80)
Standard deviation = sqrt(0.2301 / 80)
Standard deviation = sqrt(0.00287625)
Standard deviation = 0.05366 (rounded to five decimal places)
The standard deviation of the proportion of adults over 50 who wear glasses is approximately 0.05366.
To find the mean and standard deviation of those adults who wear glasses for driving in a random sample of 80 adults over 50, we will use statistical formulas.
First, let's calculate the mean or average. The mean is given by the formula:
Mean = (sum of all values) / (number of values)
In this case, we know that 63% of all adults over 50 wear glasses for driving. So, the number of adults who wear glasses in the sample is:
Number of adults who wear glasses = (63 / 100) * 80
Next, we can find the mean by dividing the number of adults who wear glasses by the total sample size:
Mean = Number of adults who wear glasses / Total sample size
Now, let's calculate the standard deviation. The standard deviation measures the dispersion or spread of the data around the mean. It is given by the formula:
Standard deviation = square root of [(sum of (values - mean)^2) / (number of values)]
However, since we don't have individual data for each adult in the sample, we need to use the information available to us. Given that 63% of all adults over 50 wear glasses, we can assume that this percentage remains the same in the sample. Hence, the standard deviation in this case can be approximated as:
Standard deviation ≈ square root of [(percentage of adults who wear glasses) * (percentage of adults who do not wear glasses) * (total sample size)]
Thus, the standard deviation is given by:
Standard deviation ≈ square root of [(63 / 100) * (37 / 100) * 80]
After calculating the above expressions, you will obtain the mean and approximate standard deviation for adults who wear glasses for driving in a random sample of 80 adults over 50.