Question

The median number of part-time employees at fast-food restaurants in a particular city was known to be 18 last year. City officials think the use of part-time employees may be increasing. A sample of nine fast-food restaurants showed that seven restaurants were employing more than 18 part-time employees, one restaurant was employing exactly 18 part-time employees, and one restaurant was employing fewer than 18 part-time employees. Can it be concluded that the median number of part-time employees has increased

Answer 1

Answer:

It can be concluded that the median number of part-time employees has increased hence we will reject the null hypothesis ( H0 : p = 0.5 )

Step-by-step explanation:

Test using  α = 0.5 to determine whether the median number of part-time employees has increased

number of restaurants with more than 18 part-time employees = 7  ( + sign )

number of restaurants with less than 18 part-time employees = 1 ( - sign )

number of restaurants with exactly 18 part-time employees = 1

first step : ( state the null and alternate hypothesis )

Null hypothesis : ( H0) : median ≤ 18

Alternate hypothesis : ( Ha ) : median ≥ 18

The size of the sample ( n ) can be considered to be 8 because

number of restaurants with more than 18 part-time employees = 7  ( + sign )

number of restaurants with less than 18 part-time employees = 1 ( - sign )

Hence the actual hypothesis that should be tested will be :

H0 : p = 0.5

Ha : p ≠ 0.5

Next apply the binomial distribution to determine the number of + signs

= nP = 8 ( 0.5 ) = 4 + signs ( right tailed test i.e. upper tail of the binomial distribution )

determine the P ( ≥ 7 ) + signs in order to obtain the p-value of this right tailed test ( using the binomial probability table )

P ( ≥ 7 )+ signs  = p(7) +signs  + p(8)+signs

                         = 0.0313 + 0.0039 = 0.0352

Hence the P-value = 0.0352  is < 0.05  hence we will reject the Null hypothesis ( H0 : p = 0.5 )

hence It can be concluded that the median number of part-time employees has increased

Answer 2

The true statement is that, the median number of part-time employees has increased

The given parameters are:

• $\mathbf{n_1 = +7}$ ---- restaurants with more than 18 part-time employees
• $\mathbf{n_2 = -1}$ ---- restaurants with less than 18 part-time employees
• $\mathbf{n_3 = 1}$ ---- restaurants with exactly 18 part-time employees

Using a 0.5 test of significance, the null and the alternate hypotheses are:

• Null hypothesis : $\mathbf{H_0 : p= 0.5}$
• Alternate hypothesis : $\mathbf{H_a : p \ne 0.5}$

The sample size (n) is calculated using

$\mathbf{n =n_1 -n_3}$

So, we have:

$\mathbf{n =7 --1}$

$\mathbf{n =8}$

The mean of the distribution is:

$\mathbf{\bar x = np}$

This gives

$\mathbf{\bar x = 8 \times 0.5}$

$\mathbf{\bar x = 4}$

Using the right tailed test, we calculate the probability that a restaurant has more than 7 part-time employees.

This is calculated as:

$\mathbf{P(x \ge 7+) = P(x = 7+) + P(x = 8+)}$

Using the binomial probability table, we have:

$\mathbf{P(x \ge 7+) = 0.0313 + 0.0039 }$

$\mathbf{P(x \ge 7+) = 0.0352}$

By comparison,

When the p-value is less than the level of significance, then we will reject the Null hypothesis

Hence, it can be concluded that the median number of part-time employees has increased

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